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GlU^l^ 




MATHEMATICAL MONOGRAPHS 

EDITED BY 

MANSFIELD MERRIMAN and ROBERT S. WOODWARD 



No. 14 

ALGEBRAIC 
INVARIANTS 



BY 

LEONARD EUGENE DICKSON 

Professor of Mathematics in the University of Chicago 



FIRST EDITION 

FIRST THOUSAND 



NEW YORK 

JOHN WILEY & SONS, Inc. 

London: CHAPMAN & HALL, Limited 

1914 



Copyright, 1914, 



LEONARD EUGENE DICKSON 



V\^xw 



THE SCIENTIFIC PRESS 

ROBERT DRIMMOND AND COMPANY 
BROOKLYN, N. V. 



EDITORS' PREFACE. 



The volume called Higher Mathematics, the third edition 
of which was published in 1900, contained eleven chapters by 
eleven authors, each chapter being independent of the others, 
but all supposing the reader to have at least a mathematical 
training equivalent to that given in classical and engineering 
colleges. The publication of that volume was discontinued in 
1906, and the chapters have since been issued in separate 
Monographs, they being generally enlarged by additional 
articles or appendices which either amplify the former pres- 
entation or record recent advances. This plan of publication 
was arranged in order to meet the demand of teachers and 
the convenience of classes, and it was also thought that it 
would prove advantageous to readers in special lines of mathe- 
matical literature. 

It is the intention of the publishers and editors to add other 
monographs to the series from time to time, if the demand 
seems to warrant it. Among the topics which are under con- 
sideration are those of elliptic functions, the theory of quantics, 
the group theory, the calculus of variations, and non-Euclidean 
geometry; possibly also monographs on branches of astronomy, 
mechanics, and mathematical physics may be included. It is 
the hope of the editors that this Series of Monographs may 
tend to promote mathematical study and research over a wider 
field than that which the former volume has occupied. 



PREFACE 



This introduction to the classical theory of invariants of 
algebraic forms is divided into three parts of approximately 
equal length. 

Part I treats of linear transformations both from the stand- 
point of a change of the two points of reference or the triangle 
of reference used in the definition of the homogeneous coor- 
dinates of points in a line or plane, and also from the stand- 
point of projective geometry. Examples are given of invariants 
of forms / of low degrees in two or three variables, and the 
vanishing of an invariant of / is shown to give a geometrical 
property of the locus / = 0, which, on the one hand, is inde- 
pendent of the points of reference or triangle of reference, 
and, on the other hand, is unchanged by projection. Certain 
covariants such as Jacobians and Hessians are discussed and 
their algebraic and geometrical interpretations given; in 
particular, the use of the Hessian in the solution of a cubic 
equation and in the discussion of the points of inflexion of 
a plane cubic curve. In brief, beginning with ample illustra- 
tions from plane analytics, the reader is led by easy stages 
to the standpoint of hnear transformations, their invariants 
and interpretations, employed in analytic projective geometry 
and modern algebra. 

Part II treats of the algebraic properties of invariants 
and covariants, chiefly of binary forms: homogeneity, weight, 
annihilators, seminvariant leaders of covariants, law of reciproc- 
ity, fundamental systems, properties as functions of the roots, 
and production by means of differential operators. Any 
quartic equation is solved by reducing it to a canonical form 
by means of the Hessian (§33). Irrational invariants are 
illustrated by a carefully selected set of exercises (§ 35). 

V 



vi PREFACE 

Part III gives an introduction to the symbolic notation 
of Aronhold and Clebsch. The notation is first explained at 
length for a simple case; Hkewise the fundamental theorem 
on the types of symbolic factors of a term of a covariant of 
binary forms is first proved for a simple example by the method 
later used for the general theorem. In \dew of these and 
similar attentions to the needs of those making their first 
acquaintance with the symboHc notation, the difiiculties usually 
encountered will, it is beHeved, be largely avoided. This 
notation must be mastered by those who would go deeply 
into the theory of invariants and its apphcations. 

Hubert's theorem on the expression of the forms of a set 
linearly in terms of a finite number of forms of the set is proved 
and appHed to estabHsh the finiteness of a fundamental set 
of covariants of a system of binary forms. The theory of 
transvectants is developed as far as needed in the discussion 
of apolarity of binary forms and its appHcation to rational 
curves (§§ 53-57), and in the determination by induction of 
a fundamental system of covariants of a binary form without 
the aid of the more technical supplementary concepts employed 
by Gordan. Finally, there is a discussion of the t}'pes of s>Tn- 
boHc factors in any term of a concomitant of a system of 
forms in three or four variables, with remarks on line and plane 
coordinates. 

For further developments reference is made at appropriate 
places to the texts in EngUsh by Salmon, EUiott, and Grace 
and Young, as well as to Gordan's Invariantentheorie. The 
standard work on the geometrical side of invariants is Clebsch- 
Lindemann, Vorlesungen iiber Geometrie. Reference may be 
made to books by W. F. Meyer, Apolaritdt iind Rationale Curve, 
Bericht iiber den gegenwarligen Stand der Invariantentheorie, and 
FormentJieorie. Concerning invariant-factors, elementary divi- 
sors, and pairs of quadratic or bilinear forms, not treated here, 
see Muth, Elementartheiler , Bromwich, Quadratic Forms and 
their Classification by Means of Invariant Factors, and Bocher's 
Introduction to Higher Algebra. Lack of space prevents also 
the discussion of the invariants and covariants arising in the 



PREFACE vii 

theory of numbers; but an elementary exposition is available 
in the author's recent book, On Invaria?its and the Theory of 
Numbers, pubKshed, together with Osgood's lectures on func- 
tions of several complex variables, by the American Mathematical 
Society, as The Madison Colloquium. 

In addition to numerous illustrativ^e examples, there are four- 
teen sets of exercises which were carefully selected on the basis 
of experience with classes in this subject. 

The author is indebted to Professor H. S. White for suggest- 
ing certain additions to the initial list of topics and for reading 
the proofs of Part I. 
Chicago, May, 1914. 



TABLE OF CONTENTS 



PART I 

Illustrations, Geometrical Interpretations and Applications 
OF Invariants and Covariants 

PAGE 

§ 1. Illustrations from Plane Analytics 1 

§ 2. Projective Transformations 4 

§ 3. Homogeneous Coordinates of a Point in a Line 8 

§ 4. Examples of Invariants 9 

§ 5. Examples of Covariants 11 

§ 6. Forms and Their Classification 14 

§ 7. Definition of Invariants and Covariants 14 

Exercises 15 

§ 8. Invariants of Covariants 16 

§ 9. Canonical Form of a Binary Cubic: Solution of Cubic Equations 17 

§ 10. Covariants of Covariants 18 

§ 11. Intermediate Invariants and Covariants 19 

Exercises 20 

§ 12. Homogeneous Coordinates of Points in a Plane 20 

§ 13. Properties of the Hessian 23 

§ 14. Inflexion Points and Invariants of a Cubic Curve 26 

Exercises 28 

PART II 

Theory of Invariants in Non-symbolic Notation 

§,15. Homogeneity of Invariants 30 

§ 16. Weight of an Invariant of a Binary Form 31 

§ 17. Weight of an Invariant of any System of Forms 32 

Exercises 33 

§ 18. Products of Linear Transformations 33 

§ 19. Generators of all Binary Linear Transformations 34 

§ 20. Annihilator of an Invariant of a Binary Form 34 

Example and Exercises ; 36 

§ 21. Homogeneity of Covariants 37 

§ 22. Weight of a Covariant of a Binary Form 38 

§ 23. Annihilators of Covariants 39 

Exercises 40 



X TABLE OF CONTENTS 

PAGE 

§ 24. Alternants 41 

§ 25. Seminvariants as Leaders of Binary Covariants 42 

§ 26. Number of Linearly Independent Seminvariants 43 

§ 27. Hermite's Law of Reciprocity 45 

Exercises 46 

§§ 28-3L Fundamental System of Covariants 47 

§§ 32, 33. Canonical Form of Binary Quartic; Solution of Quartic Equations. . 50 

§ 34. Seminvariants in Terms of the Roots 53 

§ 35. Invariants in Terms of the Roots 54 

Exercises 55 

§ 36. Covariants in Terms of the Roots 56 

Exercises 58 

§ 37. Covariant with a Given Leader 58 

§ 38. Differential Operators Producing Covariants 5!) 

Exercises 61 

PART III 

Symbolic Notation 

§§ 39-41. The Notation and its Immediate Consequences 63 

Exercises 65, 66 

§§ 42-45. Covariants as Functions of Two Symbolic Types 67 

§ 46. Problem of Finiteness of Covariants 70 

§ 47. Reduction to Problem on Invariants 71 

§ 48. Hubert's Theorem on a Set of Forms 72 

§§ 49, 50. Finiteness of a Fundamental System ot Invariants 73 

§ 51 . Finiteness of Syzygies 76 

§ 52. Transvectants 77 

§§ 53, 54. Binary Forms Apolar to Given Forms 78 

§§ 55, 56. Rational Plane Cubic Curves 81 

§ 57. Rational Space Quartic Curves 83 

§§ 58, 59. Fundamental System of Covariants of Linear Forms; of a Quadratic 

Form ; Exercises 84 

§ 60. Theorems on Transvectants; Convolution 85 

§ 61. Irreducible Covariants Found by Induction 87 

§ 62. Fundamental System for a Binary Cubic 89 

§ 63. Results and References on Higher Binary Forms 91 

§ 64. Hermite's Law of Reciprocity Symbolically 91 

§ 65. Ternary Form in Symbolic Notation 92 

Exercises ^^ 

§§ 66, 67. Concomitants of Ternary Forms 93 

§ 68. Quaternary Forms 97 

Index 9^ 



ALGEBRAIC INVARIANTS 



PART I 



ILLUSTRATIONS, GEOMETRICAL INTERPRETATIONS AND 
APPLICATIONS OF INVARIANTS AND COVARIANTS. 

1. Illustrations from Plane Analytics. If x and y are the 
coordinates of a point in a plane referred to rectangular axes, 
while x' and y' are the coordinates of the same point referred 
to axes obtained by rotating the former axes counter-clock- 
wise through an angle 6, then 

T: x = x' cos d — y' sin 6, ;y = a;' sin ^+7' cos ^. 

Substituting these values into the linear function 

l = ax-\-by-\-c, 

we get a'x' +b'y' -\-c, where 

a' =a cos d-\-h sin 6, h' = —a sin d+h cos 6. 
It follows that 

Accordingly, a?-\-lr is called an invariant of I under every 
transformation of the type T. 

Similarly, under the transformation T let 

L = Ax^By+C = A'x'-^B'y'-\-C, 
so that 

^' = ^ cos ^+5 sin ^, B' = -Asm B+B cos 6. 



ALGEBRAIC INVARIANTS 



By the multiplication * of determinants, we get 



a' 


h' 




a h 




A' 


B' 




A B 




a' 


-h' 




a —b 




B' 


A' 




B A 





cos d —sin6 
sin d cos d 



aB-bA, 



= aA-\-bB. 



cos 6 sm 6 
-sin d cos 6 

The expressions at the right are therefore invariants of the 
pair of linear functions I and L under every transformation 
of type T. The straight lines represented by 1 = and Z, = 
are parallel if and only if aB — bA=0; they are perpendicular 
if and only if aA-\-bB = 0. Moreover, the quotient of aB — bA 
by aA-\-bB is an invariant having an interpretation; it is the 
tangent of one of the angles between the two Hues. 

As in the first example, A^+B'^is an invariant of L. Between 
our four invariants of the pair /and L the following identity 

holds: 

{aA+bBy--{-{aB-bA)^ = (a^+b'~)(A^+B^-). 

The equation of any conic is of the form 5 = 0, where 

S = ax^-i-2bxy-\-cy^-\-2kx-\- 2ly + m . 

Under the transformation T, S becomes a function of x' and 
y\ in which the part of the second degree 

F = a'x'^+2b'x'y'+c'y'^ 

is derived solely from the part of S of the second degree: 

/ = ax~ + 2bxy+cy". 

The coefficient a' of x'- is evidently obtained by replacing 
X by cos d and y by sin 6 in /, while c' is obtained by replacing 
X by — sin 6 and y. by cos d in /. It follows at once that 

a'-\-c' = a-{-c. 

Using also the value of b', we can show that 

a'c'-b'^ = ac-b^, 

* We shall always employ the rule which holds also for the multiplication 
of matrices: the element in the rth row and sth column of the product is found 
by multiplying the elements of the )-th row of the first determinant by the cor- 
responding elements of the 5th column of the second determinant, and adding the 
products. 



§1] ILLUSTRATIONS FROM ANALYTICS 3 

but a more general fact will be obtained in § 4 without tedious 
multiplications. Thus a+c and d^ac — h- are invariants of 
/j and also of S, under every transformation of type T. When 
5 = represents a real conic, not a pair of straight hues, the 
conic is an ellipse ii d>0, an hyperbola if (/<0, and a parabola 
if (^ = 0. When homogeneous coordinates are used, the classi- 
fications of conies is wholly different (§ 13). 

If X and y are the coordinates of a point referred to rectan- 
gular axes and if x' and ;-' are the coordinates of the same 
point referred to new axes through the new origin (r, s) and 
parallel to the former axes, respectivel}', then 
t: x = x'-\rr, y=y'-\-s. 

All of our former expressions which were invariant under 
the transformations T are also invariant under the new trans- 
formations t, since each letter a, b, . . . involved is invariant 
under t. But not all of our expressions are invariant under 
a larger set of transformations to be defined later. 

We shall now give an entirely different interpretation to 
the transformations T and /. Instead of considering {x, y) 
and {x\ y') to be the same point referred to different pairs 
of coordinate axes, we now regard them as different points 
referred to the same axes. In the case of /, this is accomplished 
by translating the new axes, and each point referred to them, 
in the direction from (r, s) to (0, 0) until those axes coincide 
with the initial axes. Thus any point {x, y) is translated to 
a new point {x' , y'), where 

x'=x—r, y'=y—s, 
both points being now referred to the same axes. Thus each 
point is translated through a distance Vr^-\-s^ and in a direction 
parallel to the directed hne from (0, 0) to {—r, —s). 

In the case of T, we rotate the new axes about the origin 
clockwise through angle 6 so that they now coincide with 
the initial axes. Then any point (x, y) is moved to a new point 
{x' , y') by a clockwise rotation about the origin through angle 
d. By solving the equations of T, we get 

x —X cos d-\-y sin d, y' = —x sin 0-f-y cos B. 



4 ALGEBRAIC INVARIANTS 

These rigid motions (translations, rotations, and combinations 
of them) preserve angles and distances. But the transformation 
x' = 2x^ y' = 2y is a stretching in all directions from the origin 
in the ratio 2:1; while x' = 2x, y' = y is a stretching perpen- 
dicular to the ;y-axis in each direction in the ratio 2:1. 

From the multiplicity of possible types of transformations, 
we shall select as the basis of our theory of invariants the very 
restricted set of transformations which have an interpretation 
in projective geometry and which suffice for the ordinary needs 
of algebra. 

2. Projective Transformations. All of the points on a 
straight line are said to form a range of points. Project the 




Fig. 1. 

points A, B, C, . . . oi a. range from a point V, not on their 
line, by means of a pencil of straight lines. This pencil is 
cut by a new transversal in a range ^i, Bi, Ci, . . . , said to be 
perspective with the range A, B, C, . . . . Project the points 
Ai, Bi, Ci, . . . from a new vertex v by a new pencil and cut it 
by a new transversal. The resulting range of points A', B' 
C, . . . is said to be projective with the range A, B, C, . . . 
Likewise, the range obtained by any number o.' projections 
and sections is called projective with the given range, and 



§2] PROJECTIVE TRANSFORMATIONS 5 

the one-to-one correspondence thus estabhshed between cor- 
responding points of the two ranges is called a projec- 
tivity. 

To obtain an analytic property of a projectivity, we apply 
the sine proportion to two triangles in Fig. 1 and get 

4C^sinj4FC ^C ^ sin ^FC 
AV'dnACV BV~smACV' 

From these and the formulas with D in place of C, we get 

AC^AV sin A VC AD^AT sin A VD 
BC BV' sin BVC BD BV' sin BVD' 

Hence, by division 

AC . AD _ sin A VC . sin A VD 
BC ' BD "sin BVC ' sin BVD' 

The left member is denoted by (A BCD) and is called the 
cross-ratio of the four points taken in this order. Since the 
right member depends only on the angles at V, it follows that 

{ABCD) = {AiBiCiDi), 

if .4i, . . . , Di are the intersections of the four rays by a 
second transversal. Hence if two ranges are projective, the 
cross-ratio of any four points of one range equals the cross- 
ratio of the corresponding points of the other range. 

Let each point of the line AB he determined by its dis- 
tance and direction from a fixed initial point of the line; let 
a be the resulting coordinate of A, and b, c, x those of B, 
C, D, respectively. Similarly, let A', B', C, D' have the 
coordinates a' , h' , c' , x' , referred to a fixed initial point on 
their line. Then 



(^5CZ))=— 4-^ = V47-V-w = (^'^'C'£>')• 
c— x — b c —b X —b 



Hence 



x' — b' _yX — b r,_c — a 

—, — K , k— r 



X —a x—a 



c-b ' c'-b' 



6 



ALGEBRAIC INVARIANTS 



SO that yfe is a finite constant ?^0, if C is distinct from A and 
B, and hence C distinct from A' and B'. Solving for x', we 
obtain a relation 



L: 




x' 


_«a;+/3 ^_ 


a /3 
7 5 


5^0. 




7:r+5' 




In fact, 












ci = b - 


-ka\ 




^ = ka'h-ah', 


7 = 1- 


-^, 


8 = bk-a 



If we multiply the elements of the first column of A by 5 and 
add the oroducts to the elements of the second column, we 
get 



A = 



b'-ka' b'{b-a) 
1—k b—a 



= {b-a) 



-ka' b' 
-k 1 



= k{b-a){b'-a')9^Q, 



if B and A are distinct, so that B' and A' are distinct. 

Hence a projectivity between two ranges defines a linear 
fractional transformation L between the coordinate x of a 
general point of one range and the coordinate x' of the corre- 
sponding point of the other range. The transformation is 
uniquely determined by the coordinates of three distinct points 
of one range and those of the corresponding points of the other 
range. If the ranges are on the same line and if A' = A, 
B'=^B, C'^C, then ^ = 1, a = 8, I3 = y = 0, and x' = x. Thus 
(ABCD) = {ABCD') implies D' = D. 

Conversely, if L is any given linear fractional transfor- 
mation (of determinant 5^ 0) and if each value of x is inter- 
preted as the coordinate of a point on any given straight line 
I and the value of x' determined by L as the coordinate of a 
corresponding point on any second given straight line /', the 
correspondence between the resulting two ranges is a pro- 
jectivity. This is proved as follows : 

Let A, B, C, D he the four' points of I whose respective 
coordinates are four distinct values Xi, X2, X3, X4: of x such 
that 'YXi-\-89^0. The corresponding values Xi, X2, xz, x^ of 



§2] 



PROJECTIVE TRANSFORMATIONS 



jc' determine four distinct points A', B\ C', D' of /'. For, 
if zV./, 

,_ ,^ aXx + ^ aXj + _ A{Xi — Xj) 

yXi-\-8 yXj-{-8 {yXi+8){yXj+8) 

(A'B'C'D') = ''\~^\ ^ ''\~'^\ =r'^^^^^^^^^^ = (ABCD) 

Xs —X2 X4: —X2 X3 — X2 X4: — X2 

since, if U denotes 7:^1 + 5, 

^ ^ Wi/hkl ^ vwt/ w ^ ^' 

If A' 9^ A, project the points A', B' , C, D' from any con- 
venient vertex V' on to any line ABi through A and distinct 




Fig. 2. 

from I, obtaining the points Ai=A, B\, Ci, D\ of Fig. 2. Let 
V be the intersection of BBi with CC\ and let VD\ meet / at 
P. Then 

{ABCP) = (AiBiCiDi) = {A'B'CD') = (ABCD). 

From the first and last we have P = D, as proved above. 
Holding xi, X2, xz fixed, but allowing Xi to vary, we obtain 
two projective ranges on / and /'. If ^' = yl, we use T itself 
as yl^i and see that the ranges are perspective. 



8 ALGEBRAIC INVARIANTS 

If / and /' are identical, we first project the range on /' 
on to a new line {A'B' in Fig. 2) and proceed as before. 

Any linear fractional transformation L is therefore a pro- 
jective transformation of the points of a line or of the points 
of one line into those of another line. The cross-ratio of any 
four points is invariant. 

3. Homogeneous Coordinates of a Point in a Line. They 
are introduced partly for the sake of avoiding infinite coor- 
dinates. In fact, if 7 5^0, the value -8/y of x makes x' 
infinite. We set a; = a;i/a;2, thereby defining only the ratio of 
the homogeneous coordinates xi, X2 of a point. Leta[;'=:j;iVa[;2'. 
Then, if p is a factor of proportionaHty, L may be given the 
homogeneous form 

pxi =aXi-\-^X2, px2=yxi + 8x2, a8- ^y^O. 

The nature of homogeneous coordinates of points in a 
line is brought out more clearly by a more general definition. 
We employ two fixed points A and B of the line as points of 
reference. We define the homogeneous coordinates of a point 
P of the line to be any two numbers x, y such that 

x_ AP 
y'^'pB' 

where c is a constant 9^0, the same for all points P, while 
AP is a directed segment, so that AP=—PA. We agree 
to take y = if P^B. Given P, we have the ratio of x to y. 
Conversely, given the latter ratio, we have the ratio of ^P 
to PB, as well as their sum AP+PB = AB, and hence can 
find AP and therefore locate the point P. 

Just as we obtained in plane analytics {cf. § l) the relations 
between the coordinates of the same point referred to two 
pairs of axes, so here we desire the values of x and y expressed 
in terms of the coordinates ^ and rj of the same point P referred 
to new fixed points of reference A', B' . By definition, there 
is a certain new constant ^5^0 such that 

, "pB'- 



§3] HOMOGENEOUS COORDINATES 9 

Since A'P-\-PB' =A'B\ we may replace A'P by A'B'-PB' 
and get 

p^,_ knA'B' 

Let A have the coordinates ^' , t] , referred to A\ B' . Then 
PA=PB'-AB' = PB'-^^ Jj'^-^:''}^-'^'^' 

Sirtiilarly, if B has the coordinates ^i, rji, referred to A' , B\ 
p^_ (v^i-^vi)k-A'B' 

Hence, by division, 

Since we are concerned only with the ratio of x to y, we may 
set 

Since the location of A and B with reference to A' and B^ 
is at our choice, as also the constant c (and hence r and s), 
the values of rr]' and —r^' are at our choice, likewise srjx and 
—s^\. There is, however, the restriction A 9^B, whence ij'^i j^ r)\ ^' 
Thus a change of reference points and constant multiplier c 
gives rise to a linear transformation 



X=a^-\-^T1, y^-Y^+brj, 



^0, 



of coordinates, and conversely every such transformation can 
be interpreted as the formulas for a change of reference points 
and constant multiplier. 

4. Examples of Invariants. The Unear functions 
l^ax-\-by, L=Ax-\-By 
become, under the preceding linear transformation T, 

where 

a' = aa-\-hy, b' = al3-\-b5, A' = Aa-\-By, B' = A^+BS. 



10 ALGEBRAIC INVARIANTS 

Hence the resultant of the new linear functions is 



a' 


h' 




a 


b 




a iS 


= A 


a 


b 


A' 


B' 




A 


B 




7 5 


A 


B 



and equals the product of the resultant r — aB — bA of the 
given functions by A. Since this is true for every linear 
homogeneous transformation of determinant A, we call r an 
invariant of / and L of index unity, the factor which multiplies 
r being here the first power of A. 

Employing homogeneous coordinates for points on a line, 
we see that / vanishes at the single point {b,—a) and that 
L = only at {B,—A). These two points are identical if 
and only if b : a=B : A, i.e., if r = 0. The vanishing of the 
invariant r thus indicates a geometrical property which is 
independent of the choice of the points of reference used in 
defining coordinates on the line; moreover, the property is 
not changed by a projection of this line from an outside point 
and a section by a new line. Thus r = gives a projective 
property. 

Among the present transformations T are the very special 
transformations given at the beginning of § 1. Of the four 
functions there called invariants of / and L under those special 
transformations, r alone is invariant under all of the present 
transformations. Henceforth the term invariant will be used 
only when the property of invariance holds for all linear homo- 
geneous transformations of the variables considered. 

Our next example deals with the function 

/ = ax^ + 2bxy + cy^ . 

The transformation T (end of § 3) replaces / by 

F=Ae+2B^v+Cv^ 
in which 

A =aa^-\-2bay-\-cy^, 

B = aa(3-\-b{a5-\-^j)-\-cy8, 

If the discriminant d = ac — b^oifis zero, / is the square 
of a linear function of x and v, so that the transformed function 



§4] 



EXAMPLES OF INVARIANTS 



11 



F is the square of a linear function of ^ and 77, whence the 
discriminant D = AC — B^ of F is zero. In other words, d = 
impHes D = 0. By inspection, the coefficient of —b-, the highest 
power of &, in the expansion of D is 

Thus D — A^d is a hnear function bq+r of b, where q and r are 
functions of a, c, a, /3, 7, 6. Let a sind_c remain arbitrary, but 
give to b the values Vac and —Vac in turn. Since d = 
and D = 0, we have 

0=Vac^+r, 0=— Vac^+r, 

whence f = 9 = 0, D = A^d. Thus c/ is an invariant of / of 
index 2. Another proof is as follows: 



A^d = 



a 7 




a b 




a /3 










^ 5 




b c 




7 5 






a 7 




aa-\-by 


0/3+65 




^ 


B 


^ 5 




ba-{-c 


7 


^'/J+c 


5 




B 


C 



= D. 



We just noted that d = expresses an algebraic property of 
/, that of being a perfect square. To give the related geo- 
metrical property, employ homogeneous coordinates for the 
points in a hne. Then /=0 represents two points which coin- 
cide if and only if d = 0. Thus the vanishing of the invariant 
d oi f expresses a projective property of the points represented 
by/=0. 



5. Examples of Covariants. The 


Hessian 


(named after 


Otto Hesse) of a function /(x, y) of two variables is defined 


to be 




dj d'f 




h = 


dx'^ dxdy 

dydx dy^ 




Let /become F{^, 77) under the transformation 


T: x=a^-}-^v, y 


= 7^-f677, I 


\ = 


a 
7 b 


5^0. 



12 



ALGEBRAIC INVARIANTS 



Multiplying determinants according to the rule in § 1, we have 



hA = 



M 



dxdy 






+ 5- 



dxdy a/' 



3^93' 



dxdy 9/ 





dx dx 




dy dy 



where, by T, 

(1) ,=a^^,^=^^ + ^^^ = ^, 

dx dy 9a; 9^ 9y 9^ 9^ 



w- 



dx dy dv 



By the same rule of multiplication of determinants, 

/ 9 , 9\9^ / d , d\dF 

\ dx dy/d^ \ dx dy/dv 



hA = 



dx dy) d^ \ dx dyj dv 



Applying (l) with/ replaced by dF/d^ for the first column 
and by dF/dn for the second column, we get 

9^ _9^ 

de 

d~F 



A^h = 



drid^ 



9^9r? 

9^ 

9r?2 



Hence the Hessian of the transformed function F equals the 
product of the Hessian h of the given function / by the square 
of the determinant of the linear transformation. Conse- 
quently, h is called a covariant of index 2 of /. 

For an interpretation of /z = 0, see Exs. 4, 5, § 7. In case 
/ is the quadratic function / of § 4, /f reduces to 'id, where d 
is the invariant ac — 62. 

The Junctional determinant or Jacohian (named after C. 
G. J. Jacobi) of two functions J{x, y) and g{x, y) is defined 
to be 





M 


df 


d{f,g)_ 


dx 


dy 


dix, y) 


dg 


dg 




dx 


dy 



EXAMPLES OF COVARIANTS 



13 



Let the above transformation T replace / by F{^, rj), and 
g by G(^, 7/). By means of (l), we get 



djF, G) 
d{^, v) 



9/ , a/ 
dx dy 



dx dy 



M + 6^ 

dx dy 



dx dy 



dx 

dg 

dx 



dy 

dg 

dy 



I3\_^ d(f,g) 

5 I d{x,yy 



Hence the Jacobian of / and g is a covariant of index unity of 
/ and g. For example, the Jacobian of the linear functions 
/ and Z, in § 4 is their resultant r; they are proportional if 
and only if the invariant r is zero. The last fact is an illus- 
tration of the 

Theorem. Two functions f and g of x and y are dependent 
if and only if their Jacobian is identically zero. 
First, U g = 4){f), the Jacobian of/ and g is 



9/ 

dx 

4>'{f) 



dx 



df_ 
dy 



= 0. 



Next, to prove the second or converse part of the theorem, 
let the Jacobian of / and g be identically zero. If g is a 
constant, it is a (constant) function of /. In the contrary 
case, the partial derivatives of g are not both identically zero. 
Let, for example, dg/dx be not zero identically. Consider g 
and y as new variables in place of x and >'. ThMs f=F{g,y) 
and the Jacobian is 



dFdg 


dFdg 


dg dx 


dg dy 


dg 


dg 


dx 


3y 



dy 



^ 



dF 

dy 



dx 



dy 



14 ALGEBRAIC INVARIANTS 

Hence dF/dy is identically zero, so that F does not involve 
y explicitly and is a function of g only. 

6. Forms and their Classification. A function like ax^-\-bx^y, 
every term of which is of the same total degree in x and y, 
is called homogeneous in x and y. 

A homogeneous rational integral function oi x, y, . . . is 
called a form (or quantic) m x, y, . . . . According as the 
number of variables is 1, 2, 3, . . . , or ^, the form is called 
unary, binary, ternary, . . . , or q-ary, respectively. Accord- 
, ing as the form is of the first, second, third, fourth, . . . , or 
p\h. order in the variables, it is called linear, quadratic, cubic, 
quartic, . . . , or p-ic, respectively. 

For the present we shall deal with binary forms. It is 
found to be advantageous to prefix binomial coefficients to the 
Hteral coefficients of the form, as in the binary quadratic and 
quartic forms 

ax^-{-2bxy-{-cy^, aox^-\-4:aix^y-^Qa2X^y^-}-4:a3xy^-\-a4y^. 

7. Definition of Invariants and Covariants of Binary Forms. 

Let the general binary form / of order p, 

aox^+paix^-'y+^^^(^a2X^-y-^. . .-\-a,y^, 
be replaced by 

Ao^^+pAie-'v-\-^^~Y^A2e-'v'-\-- . .+^.^^ 

by the transformation T (§5) of determinant A?^0. If, for 
every such transformation, a polynomial I{ao, . . . , ap) has 
the property that 

7(^0, . . . , ^p)=AV(ao, . . . , ap), 

identically in ao, • • • , «p, after the ^'s have been replaced 
by their values in terms of the a's, then I{ao, . . . , ap) is 
called an invariant of index X of the form/. 



§7] DEFINITION OF INVARIANTS 15 

If, for every linear transformation T of determinant A?^0, 
a polynomial K in the coefficients and variables in / is such 
that * 

K{Aq, . . . ,Ay\ ^, r])^A^K(ao, . . . ,ap; X, y), 

identically in ao, . . . , ap, ^, -q, after the ^'s have been replaced 
by their values in terms of the a's, and after x and y have 
been replaced by their values in terms of ^ and rj from T, then 
K is called a covariant of index X of/. 

The definitions of invariants and covariants of several 
binary forms are similar. 

These definitions are illustrated by the examples in §§4, 5. 
Note that / itself is a covariant of index zero of /; also that 
invariants are covariants of order zero. 

EXERCISES 

1. The Jacobian oi J=ax--\-2bxy+cy'^ and L = rx-\-sy is 

J = 2{as-br)x-\-2{bs-cr)y. 

If J is identically zero, f=tL'^, where t is a constant. How does this 
illustrate the last result in § 5? Next, let / be not identically zero. Let 
k and / be the values of x/y for which /=0; m that for which L = and n 
that for which 7 = 0. Prove that the cross-ratio (k, tn, I, n)= —1. Thus 
the points represented by/=0 are separated harmonically by those repre- 
sented by Z,=0, /=0. 

2. If / is the Jacobian of two binary quadratic forms /and g, the points 
represented by 7 = separate harmonically those represented by /=0 
and also those represented by g = 0. Thus 7 = represents the pair of 
double points of the involution defined by the pairs of points represented 
by/=Oandg = 0. 

3. If /(x, y) is a binary form of order n, then (Euler) 

X — +y — = nf. 

dx ^dy ^ 

Hint : Prove this for /= ax^y" ~ ^ and for /= fi +f2. 

4. The Hessian of (ax+by)"' is identically zero. 
Hint : It is sufficient to prove this for x^. Why? 

* The factor can be shown to be a power of A if it is merely assumed to be 
a function only of the coefficients of the transformation. 



16 ALGEBRAIC INVARIANTS 

5. Conversely, if the Hessian of a binary form/(x, y) of order n is iden- 
tically zero, / is the nXh. power of a linear function. 

Hints: The Hessian of / is the Jacobian of dj/ dx, dj/ dy- By the 
last result in § 5, these derivatives are dependent: 

cx , dy 

where a and b are constants. Solving this with Euler's relation in Ex. 3, 
we get 

{ax-^hy) -^ = naf, {ax+by) — = nbf, 

dx dy 

9 log/_ na aJog/_ nb 
dx ax+by dy ax+by' 
Integrating, 

log;-;? log iax+by) = <i>iy)^^p{x). 

Hence (^ = 1// = constant, say log c. Thus f=c(ax + by)^. 

8. Invariants of Covariants. The binary cubic form 

( 1 ) jXx ,y)= ax^ + Zhx-y + 'icxy^ + dy^ 
has as a covariant of index 2 its Hessian 36 h: 

(2) h = rx^ + 2sxy + ty'^, r = ac-b~, 2s = ad-bc, t = bd-c^. 
Under any linear transformation of determinant A, let /become 

(3) F = A^^+-Wey}+^Cir^^+Dr^^. 

Let H denote the Hessian of F. Then the covariance of h gives 

(4) H = Re+'^S^r,-\-Tri^~=A^h, R = AC-B\ . . . 

Hence A^r, 2A~s, A-t are the coefficients of a binary quadratic 
form which our transformation replaces by one with the coeffi- 
cients R, 25, T. Since the discriminant of a binary quad- 
ratic form is an invariant of index 2, 

RT-S^ = AHA^r-AH-{A^sf\=A^{rt-s^). 

Hence rt — s^ is an invariant of index 6 of/. 

A Kke method of proof shows that any invariant of a covariant 
of a system of forms is an invariant of the forms. 



§9] CANONICAL FORM OF CUBIC 17 

As an example in the use of the concepts invariants and 
covariants in demonstrations, we shall prove that the invariant * 

(5) - 4(r/ - s^) = {ad - hcf - 4(ac - h^) {hd - c^) 

is zero if and only \i J{x/y, l)=0 has a multiple root, i.e., if 
J{x, y) is divisible by the square of a linear function of x and 
y. If the latter be the case, we can transform / into a form 
(3) with the factor ^^] then C = D = and the function (5) 
written, in capitals is zero, so that the invariant (5) itself is 
zero. Conversely, if (5) is zero, /=0 has a multiple root. 
For, the Hessian (2) is then a perfect square and hence can 
be transformed into ^^, which, by the covariance of h, differs 
only by a constant factor from the Hessian R^- of the trans- 
formed cubic (3) . Thus6' = r = 0. IfZ) = 0, thenC = 0(byr = 0) 
and (3) has the factor ^^, as affirmed. If D^O, 

9. Canonical Form of a Binary Cubic; Solution of Cubic 
Equations. We shall prove that every binary cubic form whose 
discriminant is not zero f can be transformed into X'^-\-Y^. 

For, if the discriminant (5) of the binary cubic (1) is not 
2ero, the Hessian (2) is the product of two linear functions which 
are linearly independent. Hence the cubic form / can be 
transformed into a form F whose Hessian (4) reduces to 2S^r], 
and hence has i? = 0, T = Q, 5?^0. If C = 0, then 5 = (by 
i? = 0) and/^ = .4^3^Z)r7^^Z)?^0 {hy S^Q). Taking 

^ = ^-^Y, r]=D-'Y, 

we get F = X^-\-Y^, as desired. The remaining case Cj^O 
is readily excluded; for, then Bp^O (by r = 0) and 

A=~, D = ^, AD = BC, 5 = 0. 
C B 

* It is often called the discriminant of/. It equals — a*P/27, where P is the 
product of the squares of the differences of the roots oif{x/y,\) =0. Other writers 
call a^P the discriminant of/. 

t If zero, / has a square factor and hence can be transformed into XW or X^. 



18 ALGEBRAIC INVARIANTS 

To solve a cubic equation without a multiple root, we 
have merely to introduce as new variables the factors ^ and 
7? of the Hessian. For, then, the new cubic is A |'^+Z)j?^ = 0. 

To treat an example, consider f=x^ + 6x-y + 12xy-+dy^ — 0. The Hes- 
sian is (d—8) {xy+2y^). Hence we take ^ = ^+23; and r] = y as new 
variables. We get /= ^' + (^—8)77^ If d = 9, we have ^^ + ri^ = 0, whence 
^/n= —1, — w or —co^, where w is an imaginary cube root of unity. But 
x/y+2=^/r). Hence :K;/y=— 3, — w— 2, — co2-2. 

10. Covariants of Covariants. Any covariant of a system 
of covariants of a system of forms is a covariant of the forms. 

The proof of this theorem is similar to that used in the 
following illustrations. We first show that the Jacobian of 
a binary cubic form / and its Hessian A is a covariant of index 
3 of/. We have 

d(F,H) ^^ d(f,AVi) _^, d(f,h) 
a(^, v) d{x, y) d{x, y)' 

As the second illustration we consider the forms /, L in 
Ex. 1, § 7. Their Jacobian is the double of the covariant 
K = vx-\-wy of index unity, where 

v = as — br, w = hs — cr. 

Thus K and L are covariants of the system of forms/, L. These 
two linear covariants have as an invariant their resultant 



1 = 



V w 
r s 



= as^ — 2brs-\-cr^. 



Under a linear transformation of determinant A, let / become 
A^^+. . . , and L become R^+Sr]. By the covariance of K, 

V^-{-Wv=A(.vx+wy), V=AS-BR, W = BS-CR. 

Thus our transformation replaces the hnear form having the 
coefficients A^; and Aw by one having the coefficients V and 
W. Th. resultant 

Av Aw 
r s 



§11] INTERMEDIATE INVARIANTS 19 

of this linear form and L is an invariant of index unity. Hence 



V W 
R S 



AE, 



V W 
R S 



:A2 



V w 
r s 



so that I = vs — wr is an invariant of index 2 of/ and L. 

From the earlier expression for /, we see that it is the 
resultant of / and L. We have therefore illustrated also the 
theorem that the resultant of any two binary forms is an 
invariant of those forms. 

11. Intermediate Invariants and Covariants. From the 
invariant ac — b^ of the binary quadratic form 

f=ax^-\- 2hxy + cy~ 

we may derive an invariant of the system of forms / and /' 
where 

f = a'x^ + 2b'xy+c'y'^. 

Let any linear transformation replace / and / by 

If / is any constant, the form/+// is transformed into F+tF'. 
By the invariance of the discriminant of /+//', 

{A-VtA'){C+tC')-{B+tB'Y^6?\{a+ta'){cMc')-{b+tby\, 

identically in /. The equahty of the terms free of / states 
only the known fact that ac — b^ is an invariant of/. Similarly 
the equality of the terms involving t^ states merely that 
a'c' — b"^ is an invariant of/'. But from the terms multiplied 
by i, we see that 
(1) ac'^a'c-2bb' 

is an invariant of index 2 of the system of forms /, i' . It 
is said to be the invariant intermediate between their dis- 
criminants. It was discovered by Boole in 1841. 

The method is a general one. Let K be any covariant of 
a form jix.y, . . .). Let a, 6, ... be the coefhcients of /. 
Let fix, y, . . .) be a form of the same order with the coeffi- 
cients a' , b\ . . . . If in iT we replace a by a-\-ta', b by 
b+tb', . . . , and expand in powers of /, we obtain as the 



20 ALGEBRAIC INVARIANTS 

coefficient of any power r of / a covariant of the system /, /'. 
By Taylor's theorem, this covariant is 

in which the symbolic rth power of 9/ 9a is to be replaced 
by d'/da', etc. 

EXERCISES 

1. For r=l, K^ac — b\ (2) becomes (1). 

2. Taking as K the Hessian (2) of cubic (1) in § 8, obtain the covariant 

{ac'+a'c-2bb')x'' + {ad'+a'd-bc'-b'c)xy + {bd' + b'd-2cc')y^ 

of index 2 of a pair of binary cubic forms. 

3. If (1) is zero, the pair of points given by /=0 is harmonic with the 
pair given by/' = 0. 

12. Homogeneous Coordinates of Points in a Plane. Let 
Li-. aiX+bty-hct = (f = l, 2, 3) 

be any three linear equations in x, y, such that 

ai h\ ci I 

A= a2 1)2 C2 5^0. 

^3 &3 C2, I 

Interpret x and y as the Cartesian coordinates of a point 
referred to rectangular axes. Then the equations represent 
three straight lines Lt forming a triangle. Choose the sign 
before the radical in 

_aiPC-\-hiy_^tCi 

so that pi is positive for a point (x, y) inside the triangle and 
hence is the length of the perpendicular from that point to 
Li. The homogeneous (or trilinear) coordinates of a point 
{x, y) are three numbers Xi, X2, xs such that 

pXi=kipi, pX2=k2p2, pX3=k3p3, 

where ki, k2, ks are constants, the same for all points. In 
view of the undetermined common factor p, only the ratios 
of xi, X2, X3 are defined. 



§12] 



HOMOGENEOUS COORDINATES 



21 



For example, let the triangle be an equilateral one with sides of length 
2, base on the .r-axis and vertex on the y-axis. The equations of the 
sides Li, Li, Lz are, respectively, 



Vz+'='' vr'='- "''■ 



Take each ^<=1. Then 



y + V3(x-l) y-Vz{x + \) 
pxi= ;; , pXi= , pX3=y. 



-2 



_2 



The curve XiX^^Xi"^ is evidently tangent to Z,i(i.e., Xi = 0) at Q=(010), 
and tangent to Li at P=(100). Substituting for the x< their values, we 
see that the Cartesian equation of the curve is 



ix-o 




Fig. 3. 



i{(>'-V3)^-3x=}=y2orx2+(^y+^y-|. 

Hence it is a circle with radius CP and center at the intersection C of the 
normal to L^ at P with the normal to Zi at Q. 



Changing the notation for the coeflScients of kipi, call 
them a J, hi, d. Then we have 



m 



pXi = aiX-{-biy-\-Ci, A?^0 



(^• = 1,2,3). 



22 ALGEBRAIC INVARIANTS 

Multiply the ith equation by the cof actor Ai of at in the 
determinant A and sum for i = l, 2, 3. Next use as multiplier 
the cofactor Bt of bi] finally, the cofactor d of d. We get 

Ax = p'EAiXi, Ay = pi:BiXi, A = p^CtXi. 

Hence x and y are rational functions of x\, X2, xs: 

, „. _ AiXi-{-A2X2-{-A3X3 ^ BiXi-\-B2X 2-i-B3X3 

CiXi-\-C2X2 + CsX3' CiXi-\-C2X2-hC3Xs' 

Any equation f{x, y)=0 in Cartesian coordinates becomes, 
by use of (C), a homogeneous equation (j)(xi, X2, X3)=0 in 
homogeneous coordinates. The reverse process is effected by 
use of (H). In particular, since any straight line is represented 
by an equation of the first degree in x and y, it is also rep- 
resented by a homogeneous equation of the first degree in 
Xi, X2, X3. For example, the sides of the triangle of reference 
are xi = 0, X2 = 0, X3 = 0. Conversely, any homogeneous equation 
of the first degree in xi, X2, X3 represents a straight line. 
The degree of ^ is always that of /. 

Take the y-axis as Li, the x-axis as L2, and let L3 recede to infinity by 
making C3 and 63 approach zero. Then (//) and (C) become 

Xi Xi 

pXi = x, pxi.—y, pxz=\; x=-, y=~. 

X3 X3 

We are thus led to a very special, but much used, method of passing from 
homogeneous to Cartesian coordinates and conversely. 

For a new triangle of reference, let the homogeneous coor- 
dinates of (jc, y) be yi, y2, ys- Then, as in (5'), 

pyi = a'iX^rh'iy^rc'i (i = l,2, 3). 

Inserting the values of x and y from (C), we get relations like 

i'. ryi = eax -\-f1X2 +g<X3 (^ = 1 , 2, 3) . 

Hence a change of triangle of reference and constants ki, 
k2, ks gives rise to a linear homogeneous transformation / of 
coordinates. The determinant of the coefficients in / is not 



§13] PROPERTIES OF THE HESSIAN 23 

zero, since yi=0, >'2 = 0, y3 = represent the sides of the 
new triangle. Conversely, any such transformation t may be 
interpreted as a change of triangle of reference and con- 
stants ki. 

Instead of regarding t as a set of relations between the 
coordinates of the same point referred to two triangles of 
reference, we may regard it as defining a correspondence between 
the points {xi, X2, X3) and (yi, y2, ys) of two different planes, 
each referred to any chosen triangle of reference in its plane. 
This correspondence is projective; for, it can be effected by 
a series of projections and sections, each projection being 
that of the points of a plane from a point outside of the plane 
and each section being the cutting of such a bundle of pro- 
jecting lines by a new plane. Proof will not be given here, 
nor is the theorem assumed in what follows. It is stated 
here to show that if / is any invariant of a ternary form / 
under all linear transformations /, then 7 = gives a projective 
property of the curve /=C. It is true conversely that any 
projective transformation between two planes can be effected 
by a linear homogeneous transformation on the homogeneous 
coordinates. ' Thus for three variables, just as for two (§§ 2, 3), 
the investigation of the invariants of a form under all linear 
homogeneous transformations is of e^i^ecial imoortance. 

13. Properties of the Hessian. Let f{xi, . . . , Xn) be a 
form in the independent variables xi, . . . , Xn- The Hessian 
/j of / is a determinant of order w in which the elements of 
the ith. row are 

a-/ dj ay 



dXidxi dXtdX2 dxtdXn 

Let /become <i){yi, . . . , jn) under the transformation 

T: Xi = Ciiyi+Ci2y2-\-. . .+Ci„y„ (/ = !,. . . , w), 

of determinant A = |q|. The product hA is a determinant 
of order n in which the element in the ith row and jth column 
is the sum of the products of the above elements of the ^'th 



24 ALGEBRAIC INVARIANTS 

row of h by the corresponding elements of the ^th column 
of A, and hence is 

-C\}-\-- — r C2j-\-. . .+— — Cn) 



^ a / 9/ dxi df dX2 . ■ df dxn\ ^ d 

dxXdxidyj dX2dyj ' ' ' dxndyjj dx 



d± 
dXidyj' 



= Hessian of 0. 
= 1 n 



Let a' be the determinant obtained from A by interchanging 
its rows and columns. In the product A'-M, the element 
in the rth row and jih column is therefore 

dxi dyj dXn dyj dyr dyj 

since dr is the partial derivative of Xt with respect to yr. Hence 

dyrdyj 

Thus h is Si covariant of index 2 of/. 

To make an application to conies, let / be a ternary quad- 
ratic form. Then h is an invariant called the discriminant 
of/. Let (ai, 02, as) be a point on /=0 (for example, one 
with 0:3 = 0). For Ca=ai and Ct2, Cjs chosen so that A?^0, 
transformation T makes (x) = (a) correspond to (3;) = (100). 
Hence we may assume that (100) is a point on/=0, so that 
the term in xi^ is lacking. Consider the terms xj with the 
factor xi. If /^O, / involves only X2 and xs and hence is a 
product of two linear functions, while h = 0. In the contrary 
case, we may introduce / as a new variable in place of 0:2. This 
amounts to setting l = X2, 

f = XiX2-\-aX2^-\-bX2X3-\-CX3^. 

Replacing xi by Xi — ax2 — bx3, we get :j;ia;2 — ^^3^, whose Hessian 
is 2k. Hence /=0 represents two (distinct or coincident) 
straight lines if and only if the Hessian (discriminant) of / 
is zero. 

Moreover, if the discriminant is not zero, then k^^O and we 
may replace Vkxs by X3 and get 0:1^:2— ^3^- Hence all conies, 
which do not degenerate into straight lines, are equivalent 



§ 13] PROPERTIES OF THE HESSIAN 25 

under projective transformation. If the triangle of reference 
is equilateral and the coordinates are proportional to the per- 
pendiculars upon its sides, :riX2— X3^ = is a circle (§ 12). 

On the contrary, if we employ only translations and rota- 
tions, as in plane analytics, there are infinitely many non- 
equivalent conies ; we saw in § 1 that there are then two 
invariants besides the discriminant. 

Next, to make an application to plane cubic curves, let 
f{xx, X2, xz) be a ternary cubic form. A triangle of reference 
can be chosen so that P=(001) is a point of the curve /=0. 
Then the term in x^^ is lacking, so that 

/ = ^3-/l+A-3/2+/3, 

where /« is a homogeneous function of .ri and X2 of degree /. 
We assume that P is not a singular point, so that the partial 
derivatives of / with respect to Xi, X2, and xa are not all zero 
at P. Hence /i is not identically zero and can be introduced 
as a new variable in place of xi. Thus, after a preliminary 
linear transformation, we have 

xz^xi+xz{axi~ -[-hxiX2+cx2^) +fi. 

Replace xs by xs — ^ (axi -\-bX'2) ■ We get 

F = xs^xi -\-ex3X2-+C, 

where C is a cubic function of Xi, X2, whose second partial 
derivative with respect to Xt and Xj will be denoted by Cy. 
The Hessian of F is 



H = 



If the transformation which replaced / by F is of deter- 
minant A, it replaces the Hessian k of / by H = A~h. Thus 
H = represents the same curve as h = 0, but referred to the 
same new triangle of reference as F = 0. We may therefore 
speak of a definite Hessian curve of the given curve f=0. 
In investigating the properties of these curves we may therefore 



Cn 


Ci2 


2X3 


Cv2 


C22-\-2ex3 


2eX2 


2X3 


2ex2 


2xi 



26 ALGEBRAIC INVARIANTS 

refer them to the triangle of reference for which their equations 
aren = 0,F = 0. 

The coefficient of xs^ in H is evidently — 8^. Thus P is on 
the Hessian curve if and only ii e = 0. li d is the coefficient of X2^ 
in C,xi=0 meets 7^ = at the points for which X2"{ex3-\-dx2) =0 
and these points coincide (at F) if and only if e = 0. In 
that case, P is called a point of inflexion oi F = and a-i=0 
the inflexion tangent at P. For a cubic curve /=0 witJwut a 
singular point, every point of inflexion is a point of intersection 
of the curve with its Hessian curve and conversely. 

14. Inflexion Points and Invariants of a Cubic Curve. EHm- 
inating x^ between / = 0, A = 0, we obtain a homogeneous relation 
in Xi, X2, which has therefore at least one set of solutions .t'i, X2- 
For the latter values of Xi and X2, /=0 and h = are cubic 
equations in xs with at least one common root, x's. Hence 
/=0 has at least one inflexion point {x'l, x'2, x'3). After a 
suitable linear transformation, this point becomes (001). As 
in § 13, we can transform / into F, in which e is now zero. If 
d = 0, then F = xiQ, and the derivatives 

=(J-rXi , Xi , — — —^i~ 

dxi 9x1 9X2 9x2 9x3 9x3 

all vanish at an intersection of xi=0,Q = 0. But we assume 
that there is no singular point on/=0 and thus none on F=0. 
Hence J 5^0. Replacing X2 by J~*X2, we have an F with 
d = l. Adding a multiple of Xi to X2, we get 

F=X32xi+C, C=X2=^+3iX2Xi2+aXi3, 

di Ci 



C12 C2 



^=-4X32C22 + 2Xi</., <, 

so that is the Hessian of C. By § 8, 

(f, = 3Q{ — b-xr-{-axiX2+bx2^). 
Eliminating xs^ between F = 0, ^^ = 0, we get 

Xi20 + 2C22C= 12(x2* + 6^)X22xi2+4ax2Xi3-362xi4) =0. 
If xi = 0, then X2 = and we obtain the known intersection 



§ 14] INFLEXION POINTS OF CUBIC 27 

(001). For the remaining intersections, we may set Xi = l 
and obtain from eacii root r of 

(1) r4+6&/^+4ar- 3^2 = 

two intersections (1, r, ix's). For, if x'z = ^, then C = 0, so 
that (1) would have a multiple root, whence d?'-\-W' = ^. But 
the three partial derivatives of F would then all vanish at 
{2b, ~a, 0) or (1,0,0), according as ftj^O or b = 0. Hence there 
are exactly nine distinct points of inflexion. 

For each of the four roots of (1), the three points of inflexion 
P and (1, r, ztx's) are collinear, being on X2=rxi. Since we 
may proceed with any point of inflexion as we did with P, 
we see that there are 9-4/3 or 12 lines each joining three points 
of inflexion and such that four of the lines pass through any 
one of the nine points. The six points of inflexion not on a 
fixed one of these lines therefore lie by threes on two new 
lines; three such lines form an inflexion triangle. Thus there 
are ^12 = 4 inflexion triangles. 

The fact that there are four inflexion triangles, one for 
each root r of (1), can also be seen as follows: 

iTH+rF = {rxi-X2)\x3~-rx2^-{r^-\-3b)xiX2-{r^-{-Qbr-\-3a)xi^\. 
The last factor equals 

X3'--\rx2+hir^-\-3b)xir-, 
r 

and hence is the product of two hnear functions. 

Corresponding results hold for any cubic curve /=0 without 
singular points. We have shown that / can be reduced to 
the special form 7^ by a linear transformation of a certain 
determinant A. Follow this by the transformation which 
multiplies xs by A and Xi by A"^^ and hence has the determin- 
ant A~^ Thus there is a transformation of determinant 
unity which replaces / by a form of type F, and hence replaces 
the Hessian h oi f by the Hessian H of F. Hence there are 
exactly four values of r for which i}/ = h-\-2Arf has a linear factor 
and therefore three linear factors. These r's are the roots 
of a quartic (l) in which a and b are functions of the coefi&cients 



28 ALGEBRAIC INVARIANTS 

of/. To see the nature of these functions, let Xi—\X2—nX3 
be a factor of \p. After replacing xi by 'Kx2+fJiX3 in \l/, we 
obtain a cubic function of X2 and X3 whose four coefficients 
must be zero. Eliminating X and ai, we obtain two conditions 
involving r and the coefficients of / rationally and integrally. 
The greatest common divisor of their left members is the 
required quartic function of r. Unless the coefficient of r^ is 
constant, a root would be infinite for certain /'s. The inflexion 
triangles of a general cubic curve /=0 are given by h-{-2^rf=0, 
where h is the Hessian of f and r is a root of the quartic (1) in 
which a and b are rational integral invariants of f. 

The explicit expressions for these invariants are very long; 
they are given in Salmon's Higher Plane Curves, §§ 221-2, 
and were first computed by Aronhold. For their short sym- 
bolic expressions, see § 65, Ex. 4. 

EXERCISES 

1. Using the above inflexion triangle yiy-iys — O, where 

rxi — X2 = yY, ^rXi±:{rx-i-\-kxx) = 2yi, 2yi, 
k={r^+2.b)/2, r^+k = l{r'- + h)9^Q, 

as shown by use of (1), we have the transformation 

^rXi = yn-\-yi, {r--\-k)xi = ryi+D, {r'^+Ji)x2= —kyi-\-rD, 
where D = y2—y:u Using (1) to eliminate a, show that 

^ {r".j^b)F = -{y-^-y,') +Zyxyty,--(r^+U)yiK 
8 r 8 

Adding the product of the latter by 54 to its Hessian, we get the product 
of yiyiyz by Z\r'^ +h) / r'^ . Hence the nine points of inflexion are found by 
setting yi, y^, yz equal to zero in turn. 

2. By multiplying the y's in Ex. 1 by constants, derive 

/= a(3,3 +s,3 +233) +6|3ZiZ2Z3, 

called the canonical form. Its Hessian is ^%, where 

// = -aiS^Si' +S..' +23=') + (a3 +2/33)ZiZ223. 

Thus find the nine inflexion points and show that the four inflexion triangles 
are 

ZiZ2Z3 = 0, 2Zi'— 3/2iZiZ3 = (/=!, w, w^), 



14] 



INFLEXION POINTS OF CUBIC 



29 



where w is an imaginary cube root of unity. Their left members are 
constant multiples of 3h+rf, where r = 3/32, — (/«— ^)2 are the four roots 
of (1), with 

3. The Jacobian of /i(a;i, . . . , Xn), . . . , fnixi, . . , Xn) is 



dfndfn 






Show that it is a covariant of index unity of /i, . . . , fn- 

4. Hence the resultant of three ternary linear forms is an invariant of 
index unity. 

5. If /i, . . . ,fn are dependent functions, the Jacobian is zero. 



PART II 



THEORY OF INVARIANTS IN NON-SYMBOLIC 
NOTATION 

15. Homogeneity of Invariants. We saw in § 11 that two 
binary quadratic forms / and /' have the invariants 

d = ac-b'^, s = ac'+a'c-2bh' 

of index 2. Note that s is of the first degree in the coefficients 
a, b, c oi f and also of the first degree in the coefficients of /', 
and hence is homogeneous in the coefficients of each form 
separately. The latter is also true of d, but not of the invariant 
s-\-2d. 

When an invariant of two or more forms is not homogeneous 
in the coefficients of each form separately, it is a sum of invariants 
each homogeneous in the coefficients of each form separately. 

A proof may be made similar to that used in the following 
case. Grant merely that s-\-2d is an invariant of index 2 of 
the binary quadratic forms/ and/'. In the transformed forms 
(§ 11), the coefficients A, B, C oi F are linear in a, b, c; the 
coefficients A', B' , C of F' are Hnear in a', b', c' . By hypothesis 

AC ^A'C-2BB' ^2{AC-B'^)=^\s^2d). 

The terms 2d!^ of degree 2 in a, b, c on the right arise only 
from the part 2(AC-B^) on the left. Hence d is itself an 
invariant of index 2; likewise s itself is an invariant. 

However, an invariant of a single form is always homo- 
geneous. For example, this is the case with the above dis- 
criminant d of /. We shall deduce this theorem from a more 

general one. 

30 



§15] HOMOGENEITY OF INVARIANTS 31 

Let / be an invariant of r forms /i, . . . ,/r of orders p\y 
. . ., pT in the same q variables Xi, . . . , Xq. Let a particular 
term ^ of / be of degree di in the coefficients of /i, of degree 
d2 in the coefficients of /2, etc. Apply the special transformation 

Xi=a^i, X2=(x^2, • • ., XQ=a^g, 

of determinant A =a'^. Then/t is transformed into a form whose 
coefficients are the products of those of ft by a^K Hence in 
the function / of the transformed coefficients, the term cor- 
responding to / equals the product of t by 

This factor therefore equals A^, if X is the index of the invariant. 
Thus 

r 

S dipi = \q. 

Hence I,dipi is constant for all the terms of the invariant. 

For the above two quadratic forms, r = pi — pi = 2. For invariant d, 
we have di = 2, di = 0, '^dipi = 4: = 2\. For s, we have di = d.,= l, Zdtpi = 4:. 
Again, the discriminant (§8) of the binary cubic form is of constant degree 
4 and index X = 6; we have 7:dipi = i-3 = 2\. 

If, as in the last example, we take r = l, we see that an 
invariant of index X of a single ^-ary form of order p is of 
constant degree d, where dp = X^, and hence is homogeneous. 

16. Weight of an Invariant / of a Binary Form f. Give to 

I and/ the notations in § 7. Let 

be any term of /, and call 

^ = ^1+2^2+3^3 + . . .+pep 

the weight of /. Thus w is the sum of the subscripts of the 
factors Oi each repeated as often as its exponent indicates. 
We shall prove that the various terms of an invariant of a binary 
form are of constant weight, and hence call the invariant isoharic. 
For example, aQX^-\-2a\xy-\-a2y^ has the invariant 00^2— fli^, 
each of whose terms is of weight 2. 



32 ALGEBRAIC INVARIANTS 

To prove the theorem, apply to/ the transformation 

X=^, y=ari. 

We obtain a form with the Uteral coefficients 

Ao = ao, Ai=aia, A2 = a2a^, . . . , Ap = apa^. 

Hence if I is of index X, 

/(go, aia, . . . , apaP)=a^I{ao, ai, . . . , Up), 

identically in a and the a's. The term of the left member 
which corresponds to the above term / of / is evidently 

CoQo^o . . . Qp^'va^. 

Hence w = X. The weight of an invariant of degree d oi 2. 
binary p-\z is thus its index and hence (§ 15) equals \dp. 

17. Weight of an Invariant of any System of Forms. Let 

/i, . . . , /„ be forms in the same variables X\^ . . . , Xg. We 
define the weight of the coefficient of any term of ft to be 
the exponent of Xa in that term, and the weight of a product 
of coefficients to be the sum of the weights of the factors. 
For q = 2, this definition is in accord with that in § 16, where 
the coefficient at of a:iP~%2^' was taken to be of weight k. 
Again, in a ternary quadratic form, the coefficients of xi^, 
a;iit;2 and X2^ are of weight zero, those of X1X3 and ::C2^3 of weight 
unity, and that of xs^ of weight 2. 

Under the transformation of determinant a, 

Xl = §1, . . . , Xq —I =^ ^q—l, Xg =Q;^j, 

fi becomes a form in which the coefficient c' corresponding 
to a coefficient c of weight k in ft is ca/^. If / is an invariant, 
7(c')^q:V(c), identically in a. Hence every term of I is of 
weight X. 

Thus any invariant of a single form is isoharic; any invariant 
of a system of two or more forms is isoharic on the whole, hut 
not necessarily isoharic in the coefficients of each form separately. 

The index equals the weight and is therefore an integer ^ 0. 



§18] PRODUCTS OF LINEAR TRANSFORMATIONS 33 

EXERCISES 

1. The invariant a^a' 2+a2a' Q—2aia' i of 

Gox"^ -\-2aixy+a2y~, a' ox"^ -\-2a\xy -\-a'2y'^ 

is of total weight 2, but is not of constant weight in ao, Ci, a2 alone. 

2. Verify the theorem for the Jacobian of two binary h'near forms. 

3. Verify the theorem for the Hessian of a ternary quadratic form. 

4. No binary form of odd order p has an invariant of odd degree d. 






^0, 



18. Products of Linear Transformations. The product TT' of 

a 13 
7 5 
' a' 

a p 
7 

is defined to be the transformation whose equations are obtained 
by eliminating ^ and r? between the equations of the given 
transformations. Hence 

la"=aa'+/37',i8"=a/3'+/35',7" = 7a' + 57',5" = 7/3' + 55'. 

Its determinant is seen to equal AA' and hence is not zero. 
By solving the equations which define T, we get 

,.5 /3 —7 a 

A A-^ A A-^ 

These equations define the transformation T~^ inverse to T\ 
each of the products TJ"^ and T~'^T is the identity trans- 
formation a; = X, y=Y . 

The product of transformation Tq, defined in § 1, by T^' is seen to equal 
Tgj^Q', in accord with the interpretation given there. The inverse of 
Tb is 

T -e: ^ = X cos e -\-y ?,\n e , 77= — x sin e+y cos 0. 

Consider also any third linear transformation 

Ti: X=axU+&iV, Y = yiU+biV. 

To prove that the associative law 

{Tr)Ti = T{T'Ti) 



34 ALGEBRAIC INVARIANTS 

holds, note that the first product is found by eliminating first 
^, rj and then X, Y between the equations for T, T, Ti, while 
the second product is obtained by eliminating first A"^, Y and 
then ^, 7] between the same equations. Thus the final eliminants 
must be the same in the two cases. 

Hence we may write TT'Ti for either product. 

19. Generators of All Binary Linear Transformations. Every 
binary linear homogeneous transformation is a product of the 
transformations 

Tn: x=^+nri, y = v; 

S,: x=^, y = kr) (k^O); 

V: x=-v, y=^- 

From these we obtain * 

]/'-i = F^: x = v, y= — ^', 

V-^T-nV=^T'n: x = x', y = y-\-nx'; 

V-'StV =S\: x = kx', y-^y' (k^O). 

For 55^0, the transformation T in § 18 equals the product 

For 5 = 0, SO that iSy^^O, T equals 

SyS' -pi -a/pV ' 

20. Annihilator of an Invariant of a Binary Form. The 

binary form in § 7 may be written as either of the sums 

/=S y.\ aiX^-^y = ^ y.\ ap-iX'yP-K 

Transformation V, of determinant unity, replaces the second 
sum by 

ii^^a,-t{-ly^r>-^r,i, 

Comparing this with the first sum we see that an invariant 

of /must be unaltered when 

(1) di is replaced by { — lYap-i (i = 0, 1, . . . ,p), 

* The T's are of the nature of translations, and the 5's stretchings. 



§ 20] ANNIHILATOR OF INVARIANTS 35 

By § 16, a function I{ao, ..., Gp) is invariant with respect 
to every transformation St if and only if it is isobaric. 

Finally, the function must be invariant with respect to 
every r»; under this transformation let 

Differentiating partially with respect to n, we get 

since T7 = y is free of n, while ^ = x — nri. The total coeflScient 
of ^p-V is 

the second term being absent if 7=0. But 

Hence] 

dn dAi dAo QA3 ^ QAp 

Now I(ao, . . . , dp) is invariant with respect to every 
transformation T„, of determinant unity, if and only if 

I{Ao, . . . , Ap)=I{aQ, . . . , dp), 

identically in n and the a's. This relation evidently implies 

dljAo, • • • , ^p) ^Q 

Conversely, the latter implies that /(^o, . • . , Ap) has the 
same value for all values of n and hence its value is that given 
by n = 0, viz., I(ao, . . . , Gp). Hence / has the desired property 
if and only if the right member of (2) is zero identically in 
n and the a's. But this is the case if and only if 

f2/(ao, . . . , ap)=0, 



36 ALGEBRAIC INVARIANTS 

identically in the a's, where 2 is the differential operator 

dai 3^2 das 9«p 

In other words, I must satisfy the partial differential 
equation 12/ = 0. In Sylvester's phraseology, / must be anni- 
hilated by the operator Q. 

From this section and the preceding we have the important 

Theorem. A rational integral function I of the coefficients 

of the binary form f is an invariant of f if and only if I is iso- 

baric, is unaltered by the replacement (1), and is annihilated 

by n. 

EXAMPLE 

An invariant of degree d of the binary quartic (§6) is of weight 2d 
(end of § 16). For J=l, the only possible term is kai] since 0=U{ka2) 
= 2kai, we have ^=0, For J =2, we have 

/ = raodi +saia3 +ta2'^, 

i2/= (5+4r)aoa3 + (4/+35)aia2 = 0, 

5=— 4r, i = 3r, / = r(aoa«-4aia3+3cj*). 

EXERCISES 

1. Every invariant of degree 3 of the binary quartic is the product of a 
constant by 

J = ao(i2ai-\-2aia2a3 — Goaz'^ — Oi^ai— a2^. 

2. The invariant of lowest degree of the binary cubic 

aoX^+3aiX^y+3a2xy'^+a3y^ 
is its discriminant (aoOs— oiOa)^— 4(ao02— ai^)(aia3— 02^). 

3. An invariant of two or more binary forms 

C0XP1+. . ., boxP^+. . ., CoxP^+. . . 
is annihilated by the operator 

sn^oo— +2a,— +. . .+bo~-+2bi~+. . .+Co— +. . . . 
dai da2 dbi dbz dCi 

4. Every invariant of 

aox^+2aixy+a2y^, boX^+2bixy+b2y^ 



§21] HOMOGENEITY OF COVARIANTS 37 

of the first degree in the a's and first degree in the 6's is a multiple of 
a<,bi+a2bo—'2aibi. 

5. A binary quadratic and quartic have no such lineo-hnear invariant. 

6. Find the invariant of partial degrees 2, 1 of a binary linear and 
a quadratic form. 

7. Find the invariant of partial degrees 1, 2 of a binary quadratic and a 
cubic form. 

8. The first two properties in the theorem of § 20 imply that / is homo- 
geneous. For, under replacement (1), any term cao% . . . Up^P of /, of 
weight w = ei+2e2+ . . . +pep, implies a term zLcao^^ai^P-i . . . ap% 
of weight w~ep-i+2ep-2+ . . . + ip — l)ei+ peo. Adding the two 
expressions for w, show that the degree d = ea+ei+ . . . -{-Cp is the constant 
2w/p. 

21. Homogeneity of Covariants. A covariant which is not 
ho}}ws;cncous in the variables is a sum of covariants each homo- 
geneous in the variables. 

For, il a, b, . . . are the coefficients of the forms, and K 
is a covariant, 

K(A, B,. . .; ^,v, ■ • .)=A^K(a, b, . . .; x, y, . . .). 

When X, y, . . . are replaced by their linear expressions in 
I, 77, ... , the terms of order co in x, y, . . . on the right (and 
only such terms) give rise to terms of order co in ^, 7?, . . . on 
the left. Hence, if A'l is the sum of all of the terms of order 
0) of K, 

Ki(A, B,...;^,v,.. .)=A^Ki(a, b, . . . ; x, y, . . .), 

and A'l is a covariant. In this way, K = Ki-\-K2 + . . . . 

Henceforth, we shall restrict attention to covariants which 
are homogeneous in the variables, and hence of constant order. 

A covariant K of constant order oi of a single form f is homo- 
geneous in the coefficients, and hence of constant degree d. 

For, let / have the coefficients a, b, . . . and order p, and 
apply the transformation x = a^, y=aT], .... The coefficients 
of the resulting form are A =aPa, B=a^b Thus 

K(a^a,a^'b,. . . ; a-^x,a-^y,. . .)^(aiyK(a,b, . . . ;x,y, . . .)^ 

identically in a, a, b, . . . , x, y, . . . , since the left member 



38 ALGEBRAIC INVARIANTS 

equals K(A, B, . . . ; ^, v, - - ■)- Now K is homogeneous 
in X, y, . . . , of order w; thus 

a-"K(aPa, aPb, . . .; X, y, . . .)=a^^K{a, b, . . . ; x, y, . . .). 

Thus if A' has a term of degree d in a, b, . . . , then 



a-"-aP'^=a«\ 



pd — (x} = q\, 



so that d is the same for all terms of A'. 

// / is a form of order p in q variables and if K is a covariant 
of degree d, order co and index X, then pd — w = q\. 

22. Weight of a Covariant of a Binary Form. In 

f = aQxTP-\-paixP-^y-\-. . .-\-i.\aiX^-'y^+. . .+apyP 

the weight of at is k. We now attribute the weight 1 to :«; 
and the weight to y, so that every term of / is of total 
weight p. 

Apply to/ the transformation x=^, y=ar]. The hteral 
coefficients of the resulting form are 

^0 = ^0, Ai=aai, . . ., Ap=aPap. 
If A is a covariant of degree d, order co, and index X, then 

K(Ao, . . . , Ap] ^, 7]) =a^K{ao, . . . , ap-, x, y). 

Any term on the left is of the form 

cAo'oAi'i . . . Ap'p^^yf {eo-\-ei+ . . . +ep=d). 

This equals 

coQ^mi^ . . . ap'vx'y'^-'a^-'' {W = r-\-ei+2e2+ . . . +pep). 

This must equal a term of the right member, so that 
W—<ji = \. But W is the total weight of that term. Hence 
every term of A is of the same total weight. A covariant 
of index X and order oj of a binary form is isobaric and its weight 
is co+X. 

For a form /of order p'm q variables, we attribute the weight 1 to Xi, Xi, 
. . . , ajj.i and the weight to Xq\ then (§ 17) every term of/ is of total 
weight p. By a proof similar to the above, a covariant of index X and order 
to of / is isobaric and its weight is w+X. 



§23] ANNIHILATORS OF CO VARIANTS 39 

Consider a covariant K homogeneous and of total order w in the variables 
Xi, . . . , Xq of two or more forms /j. As in § 15, K need not be homo- 
geneous in the coefificients of each form separately, but is a sum of co variants 
homogeneous in the coefificients of each. Let such a. K he oi degree dt in 
the coefificients of /(, of order />j. As in ^21, ^pidt—u — qX. The total 
weight oi K is o} + \. 

For example, ii pi = po = q = 2, 

/i = aoX^+2aixy +a2y-, fi = boX^+2bixy+biy'^. 

The Jacobian of /i and fi is 4^", where 

K = {aohi — aibo)x^ + (aob2— a2bo}xy + (aib2 — a2bi)y''. 
Here 

di = d2=l, w = 2, X = 1 , and K is of weight 3. 

23. Annihilators of Covariants K of a Binary Form. Pro- 
ceeding as in § 20, we have instead of (2) 

9 AY /I A ■ t \ V ^KdAj dK d^.dKdv 

— K{Ao, . . . , Ap-y ^, v) = 2 — 

dn j=odAj dn d^ dn dv dn 

and obtain the follov^^ing result: K is covariant with respect 
to every transformation x=^ + «?7, >' = ??, if and only if it is 
annihilated by * 

(1) fi-3'l- ffi = ao-^+. . .^pa^-x^ 

9^ \ 9a 1 9«p 

The binary form is unaltered if we interchange x and y, 
Gi and Qp-i ior i = 0, 1, . . . , p. Hence A' is covariant with 
respect to every transformation x=^, y = r}-\-fi^, if and only 
if it is annihilated by 

(2) 0-x^ (o^pa,^~Hp-l)a2^-+. • •+«.^). 

dy \ dao dai dcip-\/ 

Denote a covariant of order co of the binary p-'ic by 

K = Sx"+Six''-'^y-\-. . .+5.y'*'. 
* For another derivation, see the corollary in § 47. 



40 ALGEBRAIC INVARIANTS 

By operating on K by (2), we must have 
{OS-Si)x''+{OSi-2S'2)x^-'y^. . . + (06',_] -co^Jx^-^ 

•1.1,- XT r-u +bS^r^O, 

identically in x, y. Hence A becomes 

(3) K = Sx"-hOSx'^-'y-\-hO^Sx''-Y + - • •+^0"^/', 
whik.by O6'. = 0, 

(4) O"+'6' = 0. 

Hence a covariant is uniquely determined by its leader S. 
(Cf. §25). 

Similarly, K is annihilated by (l) if and only if 

(5) 125 = 0, nSi = uS, 1252=(a;-l)5i, ..., QS^ = S^-i. 

The function 5 of ao, • • • , dp must be homogeneous and 
isobaric (§§21, 22). If such a function 6* is annihilated by 
Q, it is called a seminvariant. If we have S^, we may find 
6*0,-1 by (5), then 8^-2, • • • , and finally Si. But if K is 
a covariant, we can derive S^ from S. For, by § 20, the 
transformation x=—-q, y=i replaces / by a form in which 
Ai = { — '[yap-u by the covariance of A', 

6'U)r + - • .=6U)r + . . .^S{a)x'^-\-. . .+6'.(a)r, 

so that Soi{a) =S{A). Hence S^ is derived from 5 by the 
replacement (1) in § 20. 

When the seminvariant leader S is given, and hence also co 
(see Ex. 1), the function (3) is actually a covariant of/; likewise 
the function whose coefficients are given by (5). Proof will be 
made in § 25. In the following exercises, indirect verification 
of the covariance is indicated. 

EXERCISES 

1. The weight of the leader 5 of a covariant of order w of a binary form 
/ is W ~u = X and hence (§ 21) is ^(pd—w). Thus S and / determine u. 

2. The binary cubic has the seminvariant S = aoa2 — ai''. A covariant 
with 5 as leader of is order to = 2 and is 

(aoflo— Ci^)^^ + (aofls — aia2)xy + (aiOs — a2-)y''. 

Since this is the Hessian of the cubic, it is a covariant. 



§ 24] ALTERNANTS 41 

3. Find the covariant of the binary cubic / whose leader is 
Co^fls— 3aoaitX2+2ai^ the only seminvariant of weight 3 and degree 3. It 
is the Jacobian of / and its Hessian. 

4. A covariant of two or more binary forms is annihilated by 

2fi-?/— , ZO-x—. 

dx dy 

5. Find a seminvariant of weight 2 and partial degrees 1, 1 of a binary 
quadratic and cubic. Show that it is the leader of the covariant 

{aob2~2aibi +a-,bo)x + (aobi — 2ai&2 +a2bi)y. 

24. Alternants. Consider the annihilators 

p p. p-i p) 

fi = 2 7(7^-1— =2 {k + l)at: 



3=1 d(Jj k=o dOk + i 

= 2 {p-j-^i)aj-^^z' {p-k)a,+^:^ 
of invariants of a binary form. We have 

P f p, p-l ?^2 1 

nO = ^ jaj-A(p-j + l)-^—^^ (p-k)a, + r-^\., 

P-I f p. P o2 

00 = 2 (p-k)at + i\(k-\-\)^ — 1-2 jaj-i-^ — 
A.-=o [ 9at+i y=i 9d't9(/> 

The terms involving second derivatives are identical. Hence 

fiO-OS2 = 2 (i-\-l)(p-i)ai—-Zi{p-i-hl)ai^ 
i=o 9^1 »=i 9^ 

= 2 (p-2i)ai--, 
t=o 9«t 

since the first sum is the first sum in fiO with j replaced by 
i-\-l, and the second is the first sum in 00 with k replaced 
by i—1. 

If 6* is a homogeneous function of ao, . . . , Op of total degree 
d and hence a sum of terms 

cao'oai^i . . . Qp'v {eo-\-ei-\-. . .+ep = d), 

we readily verify Euler's theorem: 

«=o ddi 



42 ALGEBRAIC INVARIANTS 

If 5 is isobaric, It is a sum of terms 

t = cao<^oai^i . . . dp^p (^1+2^2+. . .+pep=w) 
where w is constant; then 

P ■Ql P P g^ 

S idi — =2 iett = wt, S ?a, — = wS. 
% =0 9^1 % =0 »■ =0 9a< 

Hence if S is both homogeneous (of degree d) and isobaric 
{of weight w) in ao, . . . , dp, then 

(1) {W-0^)S = oiS, o} = pd-2w. 

A covariant with the leader S has the order w. (Ex. 1, § 23.) 
Since OS is of degree d and weight w-\-l, we have 

(1202 - 0212)5 = (120 - 012)05 +0(120 - 012)5 
= (co - 2)05 +aj05 = 2(co- 1)05. 
Hence for r = 1 and r = 2, we have 

(2) (l20'--0^12)5 = r(co-r + l)0'-i5. 
To proceed by induction, note that (2) implies 

(120^+1 -0'-+il2)5= (120'--0'-12)05+0^(12C>-Ol2)5 

= r(co-2-r+l)0'-5+coO'-5 = (r + l)(co-r)0''5, 

so that (2) holds also when r is replaced by r + 1. 

25. Seminvariants as Leaders of Binary Covariants. 

Lemma. // 5 is a seminvariant, not identically zero, of degree 
d and weight w, of a binary p-ic, then dp — 2wl0. 

Suppose on the contrary that 5 is a seminvariant for which 
w<0, where u = dp — 2w. By the definition of a seminvariant, 
125 = 0. Hence, by (2), § 24, 

(1) UO'S = r{o:-r + l)0'-''S (r = l, 2, 3, . . .) 

and no one of the coefficients on the right is zero. But 

being of degree d and weight dp + l; in fact, the largest weight 
of a function of ao, . . • , dp oi degree d is dp, the weight of 
a/. Then (1) for r = dp-wi-l gives O'*^-"'5 = 0. Then (1) 



§25] SEMIN VARIANT LEADERS OF COVARIANTS 43 

for r = dp-w gives O''^""'-' 5 = 0, etc. Finally, we get 5 = 0, 
contrary to hypothesis. 

Theorem. There exists a covariant K of a binary p-ic 
whose leader is any given seminvariant S of the p-ic. 

The covariant K is in fact given by (3), § 23. By (1), 
for r = co4-l, 

fiO"+i5 = 0. 

Hence 0""'"^5' is a seminvariant of degree d and weight 

w' =w-\-ii}-\-\ =pd—w-{-l. 

Then dp-2w' = -{pd-2w)-2 is negative. Hence (4), §23, 
follows from the Lemma. Thus K is annihilated by the 
operator (2), § 23. Next, in 



^-y^]K, 



the coefficient of x" y is 



r 



\w'S—-^U-r+\)0^-'S^-.W^S-r{u^-r+\)0:-'S\, 



which is zero by (1). Hence K is covariant with respect to all 
of the transformations Tn and T'n of § 19. Now 

r_irir_i = F: x=-y, y=x, 

as shown by eliminating ^, t], ^\, r]i between 

y= V, 1 j/ = »?i + ^i, 1 vi= y. 

Since K is of constant weight, it is covariant with respect to 
every St (§ 16). Hence, by § 19, K is covariant with respect 
to all binary linear transformations. 

26. Number of Linearly Independent Seminvariants. 

Lemma. Given any homogeneous isobaric function S of 
ao, . . . , Gp of degree d and weight w, where co = dp — 2w>0, 
we can find a homogeneous isobaric function Si of degree d and 
weight w-\-l such that fi5i =5. 



44 ALGEBRAIC INVARIANTS 

In (2), § 24, replace 5 by W'^S, whose degree is d and 
weight is w — r -}-l, so that its co is a)+2/' — 2. We get 

W^n'-'^S-0'-n'S^r{o:-\-r-l)0'-^2'-^S. 
Multiply this by 

(-1)'""^ ^ . 

^ ^ r!co(a; + l) . . . {co-\-r-l) 

The new right member cancels the second term of the new 
left member after r is replaced by r — l in the latter. Hence 
if we sum from r = l to r = w-\-l, the terms not cancelling are 
those from the first terms of the left members, that from the 
right member for r = l, and that from the second term on 
the left for r=w-]-l. But the last is zero, since O'^+^6'^0, 
ff''S being of weight zero and hence a power of ao. Hence 
we get Q,Si=S, where 

r=irlco{cjo-\-l) . . . (co+r-l) 

Theorem.* The number of linearly independent seminvariants 
of degree dand weight w of the binary p-ic is zero if pd — 2w<0, 

but is 

{w; d, p)-(w-l; d, p), 

if pd — 2w't_0, where (w; d, p) denotes the number of partitions 
of w into d integers chosen from 0, 1, . . . , p, with repetitions 
allowed. 

If p^4, (4; 2, p) = S, since 4+0, 3 + 1, 2+2 are the partitions of 4 into 
2 integers. Also, (3; 2, p) = 2, corresponding to 3+0, 2 + 1. Hence the 
theorem states that every seminvariant of degree 2 and weight 4 of the 
binary p-ic, p^i, is a numerical multiple of one such (see the Example 
in § 20). 

The literal part of any term of a seminvariant 6* specified 
in the theorem is a product of d factors chosen from ao, oi, 
. . . , Op, with repetitions allowed, such that the sum of the 
subscripts of the d factors is w. Hence there are {w; d, p) 
possible terms. Giving them arbitrary coefficients and oper- 
ating on the sum of the resulting terms with fi, we obtain 
a linear combination S' of the (w — 1; d, p) possible products 

* Stated by Cayley; proved much later by Sylvester. 



§27] LAW OF RECIPROCITY 45 

of degree d and weight w-\. By the Lemma there exists* 
an 6" for which 9.S is any assigned S' . Thus the coefficients 
of our S'^9,S are arbitrary and hence are hnearly independent 
functions of the {w; d, p) coefficients of S. Hence the con- 
dition S25 = imposes (w-1; d, p) Hnearly independent linear 
relations between the coefficients of 5 and hence determines 
(w— 1; d, p) of the coefficients of S in terms of the remaining 
coefficients. Thus the difference gives the number of arbitrary 
constants in the general seminvariant S, and hence the number 
of linearly independent seminvariants S. 

27. Hermite's Law of Reciprocity. Consider any partition 

W = «l + «2 + . . .+«5 

of IV into b^d positive integers such that p'^ni^n2 ... ^ Wj. 
Write «i dots in a row; then in a second row write m dots 
under the first no dots of the first row; then in a third row 
write «3 dots under the first ns dots of the second row, etc., 
until w dots have been written in 8 rows. 

Now count the dots by columns instead of by rows. The 
number nn of dots in the first (left-hand) column is 8; the 
number m2 in the second column isj;«i; etc. The number 
of columns is wi ^ p. Hence we have a partition 

w = mi-\-m2-\-. . .-\-m^ 
of IV into TT ^ p positive integers not exceeding d. 

Hence to every one of the {w; d, p) partitions of the first 
kind corresponds a unique one of the {w; p, d) partitions of 
the second kind. The converse is true, since we may begin 
with an arrangement in columns and read off an arrangement 
by rows. The correspondence is thus one-to-one. Hence 
(w; d, p) = {w; p, d). 

By two apphcations of this result, we get 

(w; d, p)-{w-l; d, p) = (w; p,d)-iw-l; p,d). 
Hence, by the theorem of § 26, the number of linearly independent 

♦Provided />(/-2(!ci-l)>0, which holds if /'(i-2i<:'^0. But if pd-2w<0, 
our theorem is true by the Lemma in § 25. 



46 ALGEBRAIC INVARIANTS 

seminvariants of weight w and degree d of the binary p-ic equals 
the number of weight w and degree p of the binary d-ic. 

Let dp — 2w = o}^0. Then, by the theorem of §25, each 
seminvariant in question uniquely determines a covariant of 
order w. 

The number of linearly independent covariants of degree 
d and order w of the binary p-ic equals the number of linearly 
independent covariants of degree p and order w of the binary d-ic. 

The covariants are of course invariants if and only if = 0. 

EXERCISES 

1. Show by means of (1), § 24, that w—hpd for an invariant. 

2. Show that (6; 6, 3) = 7, (5; 6, 3) = 5. Find the two linearly inde- 
pendent seminvariants of weight 6 and degree 6 of the binary cubic. 

3. There are only two linearly independent seminvariants of degree 
4 and weight 4 of a binary quartic. Find them. 

4. There is a single invariant or no invariant of degree 3 of the binary 
P-ic according as p is or is not a multiple of 4. (Cayley.) 

Hint: Every invariant of the binary cubic is a product of a constant 
by a power of its discriminant, of order 4 (§ 30). 

5. The binary p-ic has a single covariant or no covariant of order p 
and degree 2 according as p is or is not a multiple of 4. (Cayley.) 

Hint: Every covariant of the binary quadratic /is of the type c D^f*, 
where c is a constant and D the discriminant of / (§ 29.) The degree 2«+w» 
of the product equals its order 2m if m = 2n. Thus / has a covariant of 
order and degree p if and only if ^ = 4«, viz., c Z)"/"". 

6. No covariant of degree 2 has a leader of odd weight. 

7. If 5 is of degree di in the coefficients of a binary pi-ic, of degree 
d2 in the coefficients of a p2-ic, . . . , and of total weight w, (2), §24, 
holds with fi and replaced by 212 and 20, and co replaced by I,pidi—2w. 
For any such S, there exists an ^i of partial degrees di and total weight 
w + 1 for which {ZQ)Si = S. If 5 is a seminvariant, co^O. Generalize 
§§ 26, 27, using (w; du pr, di, pu . . ) to denote the number of ways in 
which w can be expressed as a sum of di or fewer positive integers ^/>i, 
of di or fewer positive integers^/»2, etc. 



§28] FUNDAMENTAL SYSTEM OF COVARIANTS 47 

Fundamental System of Covariants of a Binary Form, 

§§ 28-31 
28. Certain Seminvariants. For ao9^0, we may set 
f = aoX^-]-paixP~'^y-{-. . .+apyP = ao(x-aiy) . . . (x-apy). 
Apply to / the transformation 
r„: x=^-\-nr], y = -q. 

Then each root aj of / = is diminished by w, since 

X — aiy= ^—(ai — n)r]. 
Hence the difference of any two roots is unaltered. 

In particular, if n=—ai/ao, f is transformed into the 



reduced form 




/ . \ 














/-ao^" + ( 




w 


'2e-~n 


^ + 1 


IP] 


k'sf'-'TH 


where 






\ / 








\ f 






a' 2 = 






a'3- 


03- 


-3""'^+2''C 



and the roots of/' = are ai+oi/ao (^' = 1, • • - ^ P) Since 

fli 'Za\ (ai— ai) + . . . + («!— ofp) 
aiH = a< — = — — — , 

ao p p 

each root of /' = is a linear function of the differences of the 
roots of /=0 and hence is unaltered by every transformation 
r„. The same is true of a'-i/aa, a'^/ao, . . . , which equal 
numerical multiples of the elementary symmetric functions 
of the roots of/' = 0. Hence the polynomials 

Ao^ ao'j'-z = aoao — ar, 

A.i^ao^a'3=ao^a3 — Saoaia2-{-2ai^, 

yl4 = ao'^<i'4 = flo'^«4 — 4ao^<iia3 +600^1^02 — 3a i^* 

are homogeneous and isobaric,* and are invariants of / with 
respect to all transformations r„. By definition they are, 
therefore, seminvariants of / provided the subscript of each A 
in question does not exceed p. 

* This is evident for A2, A3, At. Further A's will not be employed here. A 
general proof follows from § 34. 



48 . ALGEBRAIC INVARIANTS 

Since /' was derived from / by a linear transformation of 
determinant unity, any semin variant S oif has the property 



S{ao, 






Hence any rational integral seminvariant is the quotient 
of a polynomial in ao, A2, . ■ . , ^j; by a power of gq. For 
/> ^ 4, we shall find which of these quotients equal rational 
integral functions of ao, . . . , Op and hence give rational integral 
seminvariants. The method is dua to Cayley. 

For p = l, S is evidently a numerical multiple of a power 
of ao. Since co is the leader of the covariant /=aox+aiy of 
/, we conclude that every covariant of a binary linear form / 
is a product of a power of / by a constant; in particular, there 
is no invariant. 

29. Binary Quadratic Form. Since A2 does not have the 
factor ao, we conclude that every rational integral seminvariant 
is a polynomial in ao and A2. Now A 2 is an invariant of / 
(§4), and ao is the leader of the covariant/ of/. Hence a 
fundamental system of rational integral covariants of the binary 
quadratic form f is given hy f and its discriminant A2. We express 
in these words our result that any such covariant is a rational 
integral function of/ and A2. 

30. Binary Cubic Form. We seek a polynomial P(ao, ^2,^3) 
with the implicit, but not expHcit, factor ao. Write A'l for 
the terms of Ai free of ao: 

(1) A'2 = -ai\ A'^ = 2a,K 

We desire that P(0, A' 2, ^'3) =0, identically in ai. Now 

4^V+^V^0, 

(2) 4.A2^+Az^^ao''D, 

where D is the discriminant of the cubic form, 

D = a^rr.:r — Gaoaia2a3 +4aoa2^ +4ai^a3 — 3ai^a2^. 



§31j FUNDAMENTAL SYSTEM OF COVAKIANTS 49 

By means of (2) we eliminate As~ and higher powers of 
A3 from P(ao, A2, A 3) and conclude that any semin variant 
is of the form w/ao'^, where tt is a polynomial in ao, A2, A3, D, 
of degree 1 or in yl 3. li k>0, we may assume that not every 
term of x has the explicit factor ao- In the latter case, t does 
not have the implicit factor ao- For, if it did, 

x' = 7r(0, A'o, A's, D')=0, D' = 4:ai^a3-Sai^a2^. 

Since as occurs in D', but not in A' 2 or ^'3, tt' is free of D'. 
By (1), the first power of A' 3 is not cancelled by a power of 
A' 2. Hence tt' is free of A' 3 and hence of A'2- 

A fundamental system of rational integral seminvariants of 
the binary cubic is given by ao, A2, A3, D. They are connected 
by the syzygy (2). 

A fundamental system of rational integral covariants of the 
binary cubic f is given by f, its discriminant D, its Hessian H, 
and the Jacobian J of f and H. They are connected by the syzygy 

(3) 4H^+J-^fW. 

The last theorem follows from the first one and (2), since 
ao, A2, A3 are the leaders of the covariants/, //, /. 

31. Binary Quartic Form. We first seek polynomials 

P(ao, A2, A3, Ai) with the impHcit, but not explicit, factor 
oq. Thus 

r = P{0,A'2,A'3,AU)^0, A'2=-ai\ A'3 = 1ax\ A\= -Za^K 

The simplest P' is evidently 3^1 '2" +.4 '4. We get 

/44+3/l22 = ao-/, / = aoa4-4aia3+3a22. 

We drop Ai, and consider polynomials 7r(ao, Ao, A3, I) with 
the imphcit, but not exphcit, factor, ao. Such a polynomial 
is given by (2) , § 30. For ao = 0, D= -ai^I = A '2I. We have 

A2l-D = aQj, 

J = aoa2a4 — aoCa^ + 2aia2a3 — ai-a4 — ao^. 
Eliminating D between this relation and (2), § 30, we get 
(1) ao-V-ao2^2/ + 4^2^+^32 = 0. 



50 ALGEBRAIC INVARIANTS 

In view of their origin, / and / are semin variants of the 
quartic /. Since they are unaltered by the replacement (1), 
§20, they are invariants of/ {cf. §20, Example and Ex. 1). 
In view of (I), tt equals a polynomial ^ in ao, A2, A^, I, /, 
of degree or 1 in ^3. Suppose that (/> does not have the 
explicit factor oq. Then the equal function of ao, . . . , a^ is 
not divisible by Qq. For, if it were, 

(/)(0, — fli^, 2ai3, 302^ — 4aia3, —ai^a^-]-. . .)— 0- 

In view of the term a^, <i> cannot involve /, and hence not /. 
Nor can </> be linear in ^3 in view of the odd power a\^. Hence 
is free of ^3 and hence of ^2. 

A fundamental system of rational integral seminvariants of 
the binary quartic is given by ao, A2, A3, I, J. They arc con- 
nected by the syzygy (1). 

A fundamental system of rational integral covariants of the 
binary quartic f is given by f, its invariants I and J, its Hessian 
H and the Jacobian G of f and H. They are connected by the 
syzygy 
(2) f^J-fmi-\-im-\-G^^O. 

The second theorem follows from the first one, since ao, 
A2, A3 are the leaders of the covariants/, H, G. 

It would be excessively laborious, if not futile, to apply 
the same method to the binary quintic, whose fundamental 
system is composed of 23 covariants,* most of which are 
very complex. The symbolic method is here superior both 
as to theory and as to compact notation (see Part III.). 

Canonical Form of Binary Quartic. Solution of Quartic 

Equations 

32. Theorem. A binary quartic form f, whose discrim- 
inant is not zero, can be transformed linearly into the canonical 
form 

(1) X^ + Y^+QmX^YK 

* Faa di Bruno, Thcorie der Bindren Formen, German tr. by Walter, 1881, 
pp. 199, 316-355. Salmon, Modern Higher Algebra, Fourth Edition, 1885, p. 
227, p. 347. 



§32] CANONICAL FORM OF QUARTIC 51 

The reason there is here a parameter m lies in the existence 
of two invariants / and / of weights (and hence indices) 4 
and 6, and hence a rational absolute invariant P/J-, i.e., one 
of index zero, and consequently having the same value for/ 
and any form derived from / by linear transformation. 

Since / vanishes for four values of x/y and hence is the 
product of four linear functions, it can be expressed (in three 
ways) as a product of two quadratic forms, say those in the 
right members of the next equations. To prove our theorem 
it suffices to show that there exist constant p, q, r, s (each 5^0) 
and a, /3 {a 9^0) such that 

p(x -]-ay)~-\-q(x-\-l3y)^ ^ ax^-\-2bxy-\-cy^, 
r(x +ay)- +5(x +/3y)- = gx^ + 2hxy + ky^. 
For, the product/ of these becomes (1) by the transformation 

A' = '^pr {x+ay) , Y = ^qs {x + ^y) , 
of determinant 5^0. The conditions for the two identities are 
p-\-q = a, pa-\-qfi = b, pa'^-\-q0^ = c, 
r-\-s = g, ra-\-s0 = k, ra^-{-s^^ = k. 
The first three equations are consistent if 

1 1 a 

a ^ b ^{^-a)=C-b(a+l3)-\-aa^ = 0. 

If p = 0, or if q = 0, the same equations give b'~ = ac, so that the 
first quadratic factor of / and hence / would have a dcuble 
root. Similarly, the last three equations have solutions r^^O, 

^-//(a+/3)+ga/3 = 0. 

If the determinant ah — bg is not zero, the last two relations 
determine a 4-/3 and a^, and hence give a and /3 as the roots of * 

{ah — bg)z- — (ak — cg)z-\-bk — ch = 0. 

* Its left member is obtained by setting x/y= — s in the Jacobian of the two 
quadratic factors of /. 



52 ALGEBRAIC INVARIANTS 

If its roots were equal, the two relations would give 

C-2ba-\-aa^ = 0, k-2ha-\-ga^ = 0, 

and the two quadratic factors of/ would vanish for x/y= -a. 

If ah-bg = 0, but ch — bk^^O, we interchange x with y 
and proceed as before. If both determinants vanish, either 
b9^0 and the second quadratic factor is the product of the 
first by h/b, or else b = and hence h = and no transfor- 
mation of/ is needed. 

33. Actual Determination of the Canonical Quartic. Let 

A denote the determinant of the coefficients of x, y in A', Y. 
Then /, its invariants / and / and Hessian H are related to 
the canonical form, its invariants and Hessian, as follows: 

/ = A4(l+3m2), J^A^(m-m^), 
Zr = A2iw(X4+F^) + (l-3w2)X2 72|_ 

Thus A^m may be found from the resolvent cubic equation 

4(A2w>^-/(A2w)+/ = o. 

Then A* may be found from /. We may select either square 
root as A2 and hence fmd m. In fact, by replacing .Y by 
XV — 1 in /, the signs of A2 and m are changed. By elim- 
inating X^+F^, we get 

If 9^2=1, / is the square of X'^dzY'^ and the discriminant of 
/ would vanish. Hence we obtain XY by a root extraction. 
Thus X and F are determined up to constant factors / and 
t'^. We may find / by comparing the coefficients of .t^ and 
x^y in / and the expansion of its canonical form, or by use 
of the Jacobian G oi f and H: 

G = ^K\-9m^~)XY{X^-Y^), 

and combining the resulting X*— F^ with the earlier A^^-f-F'*. 
Or from/ and XY we can find A'2 + F2 and then X±F. 
To solve / = 0, we have only to find the canonical form 



§34] SEMINVARIANTS IN TERMS OF THE ROOTS 53 

Seminvariants, Invariants, and Covariants of a Binary 
Form/ as Functions of the Roots of/=0, §§ 34-37. 

34. Seminvariants in Terms of the Roots. Give / the nota- 
tion used in § 28, so that ai, . . . , ap are the roots of /=0. 
After removing possible factors ^o from a given seminvariant 
of /, we obtain a seminvariant 5 not divisible by qq. Let 
5 be the degree of the homogeneous function S of the a's. 
Thus S is the product of a^ by a polynomial in ai/ciQ, . . . , dp/ao 
of degree 8. The latter equal numerical multiples of the ele- 
mentary S}'mmetric functions of ai, . . . , ap, each of which 
is linear in every root. Hence our polynomial equals a sym- 
metric polynomial a in ai, . . . , ap of degree 5 in every 
root. 

Since S is of constant weight w and since at/ao equals a 
function of total degree i in the roots, a is homogeneous in 
the roots and of total degree w in them. 

Besides being homogeneous and isobaric in the a's, a sem- 
invariant must be unaltered by every transformation Tn of 
§ 28. Under that transformation, each root is diminished, 
by w (§28). Since 

ai = ai-{-{ai — ai) (^ = 2,. . . , p) 

we can express o- as a polynomial P(ai) whose coefficients 
are rational integral functions of the differences of the roots. 
If P{ai) is of degree ^1 in ai, we have P(ai)=P{ai — tt), for 
all values of ?i. But an equation in n cannot have an infinitude 
of roots. Hence P(ai) does not involve ai, so that a equals 
a polynomial in the differences of the roots. 

Multiplying by the factors ao removed, we obtain the 
theorem : 

Any seminvariant of degree d and weight w of the binary 
form aQX^-\-. . . equals the product of a(f by a rational integral 
symmetric function a of the roots, homogeneous {of total degree 
w) in tJie roots, of degree ^ d in any one root, and expressible 
as a polynomial in the differences of the roots. 

Conversely, any such product can be expressed as a poly- 
nomial in the a's and this polynomial is a seminvariant. 



54 ALGEBRAIC INVARIANTS 

Since the factor o- is symmetric in the roots, and is of degree 
^d in any one root, its product by ao'^ equals a homogeneous 
polynomial in the a's whose degree is d. This polynomial is 
isobaric since a is homogeneous, and is unaltered by every 
transformation T„, since o- is expressible as a function of the 
differences of the roots. 

The importance of these theorems is due mainly to the 
fact that they enable us to tell by inspection (without com- 
putation by annihilators) whether or not a given function of 
the roots and ao is a seminvariant. A like remark applies to 
the theorem in § 35 on invariants and that in § 36 on covariants. 

EXAMPLE 
The binary cubic has the seminvariant 

ao*2(ai— a2)(ai — a3)=aoK^«i"~2:aia2) 
3 

= CoM(2ai)='-3Saia2}=ao-| ( — -j -sl — ) i = -9{aoa2-ai^). 

35. Invariants in Terms of the Roots. A seminvariant of 
/ is an invariant of / if and only if it is unaltered by the trans- 
formation X = - 77, >» = H§ 20) . For the latter, 

X — ay= —al ^H — r?j, 

so that ar is replaced by — I far, and hence ar-as by 

ar — as 

(XrOCs 

The coefficient of ^^ in the transformed binary form is 

Ao = ( — iyaia2 . . . ocpao. 
By § 34, any seminvariant of / is of the type 

ao'^2ci(product of w factors like ar — as). 
Hence this is an invariant if and only if it equals 

(^-iyd(^ai . . . a:p)'^ao'^Sct( product of the w corresponding-^^ 



§35] INVARIANTS IN TERMS OF THE ROOTS 55 

and hence if ±0:1'' . . . ap'^ equals the product of the factors 
arOis in the denominators. This is the case if and only if each 
root occurs exactly d times in every term of the sum and if 
pd is even. By the total number of a's, pd = 2w. 

Any invariant of degree d and weight w of the binary form 
aoX^-\- . . . equals the product of ai,^ by a sum of products of 
constants and certain dij'erences of the roots, such that each root 
occurs exactly d times in every product; moreover, tJie sum equals 
a homogeneous symmetric function of the roots of total degree w. 
Conversely, the product of any such sum by ao'^ equals a rational 
integral invariant. 

EXERCISES 

1. a(,^{ai—a2)- is an invariant of the binary quadratic form. Any 
invariant is a numerical multiple of a power of this one. 

2. ai,-2(ai— a2)'(«3— 04)' is an invariant of the binary quartic. 

3. 00^2(01— a2)(ai— a:)) is not an invariant of the binary cubic. 

3 

4. If we multiplyrtn^'''"^' by the product of the squares of the differences 
of the roots of the binary p-ic f, we obtain an invariant (discriminant of 
/) . Also verify that pd = 2w. 

5. The sum of the coefficients of any seminvariant is zero. 
Hint: Use/= (x + y)^, whose roots are all equal. 

6. Every invariant of the binary cubic is a power of its discriminant. 

7. A function which satisfies the conditions in the theorem of § 35 
except that of symmetry in the roots is called an irrational invariant. If 
aj, . . . , a4 arc the roots of a binary quartic/, and 

U=(ai—ai)(a-i—a3), V= {a> — ai)(a:i—ai) , W= {ai—a2)(a3—a4) , 

why are aou, a„v, aoW irrational invariants of /? They are the roots of 
z' — 12/2— 5 = 0, where 5- is the product of ao" by the product of the squares 
of the differences of the roots and hence is the discriminant of/. Hints: 
tt+Z)+w = 0, and s = nv-\-iiw-\-mv is a symmetric function of ai, . . . , at 
in which each aj occurs twice in every product of differences, so that ao'^s 
is an invariant of degree 2. By the Example in §20, ao^s = cI, where c 
is a constant. To determine c, take ai = l,a2= — 1, 03 = 2, «<= — 2, so that 
/=(x2-y2)(;t2-4y2), 7 = 73/12, u=-Q, v=l, w = 8, 5= -73. Hence 
c= — 12. As here, so always an irrational algebraic invariant is a root of an 
equation whose coefficients are rational invariants. 



56 ALGEBRAIC INVARIANTS 

8. If ai, oil are the roots of the binary quadratic form/, and as, a^ the 
roots of /' in § 11, the simultaneous invariant 

ac' +a'c — 2hh' = aa'lasat +aia2 — K«i +"2) (as -\-o!i) \ = \ao{u — v) , 

if the product jf is identified with the quartic in Ex. 7. Hence a simul- 
taneous invariant of the quadratic factors of a quartic is an irrational invar- 
iant of the quartic. Why a priori is the invariant three-valued? 

9. The cross-ratios of the four roots of the quartic are —v/ii, etc. These 
six are equal in sets of three if 7 = 0. For, if 5 = 0, 

, , . , —V —u —w 
vw — u\—v—w) = u^, uw = v{-'u—w) = v^, — = — — . 

U W V 

The remaining three are the reciprocals of these and are equal. 

10. By Ex. 3, § 11, one of the cross-ratios is —1 if ac'-f- . , . =0. Why 
does this agree with Ex. 8? 

11. The product of the squares of the differences of the roots of the 
cubic equation in Ex. 7 is known * to be 

— 4.(-12l)^-278'- = aoKti-v)-{n-w)-(v-wy. 

Also,* 52 = 256(7' -27/2). Hence the left member becomes 3^-4^/2. Thus 

33. 42/= zizaQ^{u—v)(u—w){v—w). 

Using J from § 31, and the special values in Ex. 7, show that the sign is 
plus. Verify that the cross-ratios equal —1, —1, 2, 2, 5, 5, if 7 = 0. 

36. Covariants in Terms of the Roots. Let K (ao, ■ . . ,ap\x, y) 
be a covariant of constant degree d (in the coefficients) and 
constant order co (in the variables) of the binary form/ = aoa;^ + . . . 
Then 

K = Qo'^fK, 

where k is a polynomial in x/y and the roots a\, . . . , ap oi 
/=0. Under the transformation Tn in §28, let / become 
^0^^+ . . . , with the roots a'l, . . . , a'p. Then 

X ? / II 
a% = cc J, aT — (Xs=OLT — OLs. 

y V 
Making use of the identities 



y \y 

Cf. Dickson, Elementary Theory of Equations, p. 33, p. 42, Ex. 7. 



§36] COVARIANTS IN TERMS OF THE ROOTS 57 

we see that k equals a polynomial P(ai) whose coefficients are 
rational integral functions of the differences of x/y, ai, . . . , ap 
in pairs. Since 

K{Ao, . . . , Ap] ^, ri)=K{ao, . . . ,ap; X, y), Ao = ao, r] = y, 
we have ^1 a'l, . . . , a'p, -j =«( ai, . . . , ap, -j. 
The left member equals -P(a'i) since 

a'i = {ai — ai)-i-a'i, -=( ai)-\-ai. 

V \y / 
Hence 

P(ai-n)-P(ai)=0 

for every n. Hence ai does not occur in P(ai), and k is a 
polynomial in the differences of x/y, ai, . . . , ap. 

Let W be the weight of K and hence of the coefficient of 
y. Then k is of total degree W in the a's and of degree co 
in x/y. Thus 

K = 2C(| product of w differences hke — aA 

[ y J 

•{product of W — o) differences like ar—as]. 
Hence 

K = ao'^ZCi\'prod\iQt of w differences like x—ary] 

• {product of W—oi differences like ar—as\. 

Next, for x=—t], y=^, f becomes F = Ao^^+ . . . with 
a root — X/ut corresponding to each root ar of /. The function 
K for F is 

product of CO differences like ^-\ — ri = - !^[ 

ar —ar \ 

• ] product of W — o) differences like — ^ [ . 

\ aras J 

Using the value of Aq in § 35, we see that the factor 

\-\yW . . . ap'^ 

must be cancelled by the —ar and the aras in the denominators. 



5S ALGEBEAIC INVARIANTS 

Thus each term of the sum involves every root exactly d times. 
The signs agree since 

as follows by counting the total number of as. 

Any covariant of degree d, order co and weight W of 

ao{x—aiy) . . . (x—apy) 

equals the product of ao^ by a sum of products of constants and 
w differences like x—ary and W — oj differences like ar—as, such 
that every root occurs in exactly d factors of each product; more- 
over, the sum equals a symmetric function of the roots. Conversely, 
the product of ao^ by any such sum equals a rational integral 
covariant. 

EXERCISES 

1. j=aoX^-\-ZaiX'^y-\-Za2xy-+a3y^ has the covariant 

K = ao'^'L{x-aiyy{oL'z-azy-. 

3 

Show that the coefficient of x"^ in K equals — 18(aoa2— Oi^). Why may we 
conclude that K= — 18H, where H is the Hessian of/? 

2. The same binary cubic has the covariant ^ 

ao^Z(x—aiy){x—a2y){a2 — a3)ia3—ai) = 9H. 
3 

3. Every rational integral covariant of the binary quadratic / is a prod- 
uct of powers of / and its discriminant by a constant. 

37. Covariant with a Given Leader S. If the seminvariant 
S has the factor ao, and S = aoQ, and if Q is the leader of a 
covariant K of /, then, since ao is the leader of /, S is the leader 
of the covariant fK. Hence it remains to consider only a 
seminvariant 6* not divisible by oq. li S is of degree d and 
weight w, 

S = ao'^XCiiproduct of w factors like ar—as), 

where each product is of degree at most d in each root, and 
of degree exactly d in at least one root (§ 34). If each product 
is of degree d in every root, S is an invariant (§ 35) and hence 
is the required covariant. In the contrary case, let a2, for 
example, enter to a degree less than d; we supply enough 
factors x—a2y to bring the degree in a2 up to d. Then ao'^ 



§38] DIFFERENTIAL OPERATORS 59 

multiplied by the sum of the total products is a covariant 
with the leader S. For example, 

ao^2(a2— 03)^, 00^2(0:2 — 0:3) («3—ai) 

3 3 

are the leaders of the covariants in Exs. 1, 2, § 36, of the binary 
cubic. The present result should be compared with the 
theorem in § 25. 

We may now give a new proof of the lemma in § 25 that 
dp — 2w^0 for any semin variant 5 of degree d and weight 
■w of the binary p-ic. Whether 5 has the factor ao or not, 
the first term of the resulting covariant K is Sx", where 
oo = dp — 2w. For, in each product in the above S, the roots 
ai, . . . , ap occur 2w times in all. In K each root occurs d 
times. Hence we inserted dp — 2w factors x—ay in deriving K 
from S. 

38. Differential Operators Producing Covariants. Let the 

transformation 

T: x = a^-\-^v, y = y^-\-8v, A=a8-^y^0 

replace /(x, )') by (/)(^ r?). Then 

d±^^dx dldy^^df^ df 
d^ dx d^ dy d^ dx dy 

d±^^dx_^^dy^^d[_^^^ 

di dx dv dy dv dx dy 
Solving, we get 

dy dv d^ dx dv d^ 

or df=D<p, dif = D]_(p, if we introduce the differential operators 

dy dx dv 9? a^ a^ 

As usual, write d'-dif for d\d{dif)]. Since the result of 
operating with d on df is the same as operating with D on 
the equal function D^ of ^ and r], we have d-f=D"4>. Similarly, 



60 



ALGEBRAIC INVARIANTS 



(r+5 = a)), 



The right member is the result of operating on with the 
operator obtained by substituting D for d/dn and Di for — d/dk 
in 

whose terms are partial derivatives of order w. Hence, if 
the form 

becomes X(^, r?) under the transformation T, our right mem- 
ber is the result of operating on ^ with \{d/drj, — d/9^). The 
left member is the result of operating on / with 



\ dy dx) • \dy dxj 



Hence if T replaces the forms J {x, y), l{x, y) by 0(^, rf), X(^, rj), 
then 



L VaV 9^/ 



0(?, ^)=A'' 



f{x, y) 



\-\dy dxj \ 
is a consequence of the equations for T, if oj is the order of l{x, y). 

Let / and / be covariants of indices m and n of one or more 
binary forms ft with the coefficients ci, C2, . . . . Under T 
let the transformed forms have the coefficients Ci, C2, .... 
Then 

/(C; ^, v) = A-/(c; x, y), 1{C; ^, v) = A«/(c; x, y). 

But 0(^, ri) =f{c\ X, y), by the earlier notation. Hence 

^{^, r})=A-^f{C; ^, v), X(^, 7;)=A-"/(C; ^, r?). 

Inserting these into the formula of the theorem, and mul- 
tiplying by A"'+'', we get 



['{''■• i'-i-^\'''-'''^^^''^^'ii'-4y'-l^) 



f{c;x,y). 



The function in the right member is therefore a covariant of 
index w+w+w of the ft. We therefore have the theorem 
of Boole, one of the first known general theorems on covariants: 



§38] DIFFERENTIAL OPERATORS 61 

Theorem. If I and J are any covariants of a system of 
binary forms, we obtain a covariant {or invariant) of the system 
of forms by operating on f with the operator obtained from I by 
replacing x by d/dy and yby— d/dx, i.e., x'"/ byf — iyd'"^^/dy''dx\ 

EXERCISES 

1. Taking l=f=ax-+2bxy+cy-, obtain the invariant 4{ac—b-) of /. 

2. If /=/ is the binary quartic, the invariant is 2 -4! / of § 31. 

3. Using the binary quartic and its Hessian, obtain the invariant /. 

4. Taking l = aoX^+. . . , f=boX^+. . ., obtain their simultaneous 
invariant 



If also /=/, we have an invariant of/, which vanishes if p is odd. For 
/> = 2 and ^ = 4, deduce the results in Exs. 1 , 2. 

5. A fundamental system of covariants of a quadratic and cubic 

Q = Ax^ +2Bxy+Cy, f= ax' +3bx-'y +3cxy^- +dy^ 

is composed of 15 forms. We may take Q and its discriminant AC—B''; 
f, its discriminant and Hessian h, given by (5) and (2) of § 8, the Jacobian 
/of/and£?: 

J=(d''d-3abc+2b')x^+3(abd+b'-c-2ac^)x'-y 

+3(2b''d-acd-bc")xy'' + {3bcd-ad'-2c')y»; 

the Jacobian of /and Q: 

{Ab-Ba)x' + i2Ac-Bb-Ca)x^y + (Ad+Bc-2Cb)xy^ + (Bd-Cc)y^; 

the Jacobian of Q and //: 

{As-Br)x^ + {At-Cr)xy + (Bt-Cs)y'-; 

the result of operating on / with the operator obtained as in the theorem 
from l=Q: 

Li = (aC+cA-2bB)x + (bC+dA-2cB)y; 
the result of operating on Q with the operator obtained from Lr. 

L2= {aBC-b{2B^+AC) +3cAB-dA']x 
+ \aO-3bBC+c(AC+2B^~)-dAB\y; 



62 ALGEBRAIC INVARIANTS 

the result Lz, of operating on / with Q and the result Li of operating on Q 
with Li (so that Li and Li may be derived from Ly and L2 by replacing 
a, . . . , d by the corresponding coefficients of J) ; the intermediate 
invariant At+Cr—2Bs of Q and h (§ 11); the resultant of Q and/: 

c2C3-6a65C2+6acC(252-.4C)+a(f(6.45C-853) +96^.4 C2 
-18bcABC+6bdA(2B^-AC)+9cW-C-()cdBA^+d\L^; 

the resultant of Zi and Z4 ( = resultant of Z,2 and L3), obtained at once as a 
determinant of order 2. Salmon, Modern Higher Algebra, § 198, gives geo- 
metrical interpretations. Hammond, Arner. Jour. Math., vol. 8, obtains the 
syzygies between the 15 covariants. 



PART III 

SYMBOLIC NOTATION 
The Notation and its Immediate Consequences, §§ 39-41 

39. Introduction. The conditions that the binary cubic 

(1) f^aoXi^-\-3aiXi"X2-{-3a2XiX2^-\-(i3X2^ 
shall be a perfect cube 

(2) (aiXi -\-a2X2y 

are found by eliminating ai and a2 between 

(3) ai'^ — Qo, ai-a2 = ai, axa2~ — CL2, a<^=az, 
and hence the conditions are 

(4) aoa2 = ai~, aia3=a2^. 

Thus only a very special form (1) is a perfect cube. 

However, in a symbolic sense * any form (1) can be rep- 
resented as a cube (2), in which ai and 0:2 are now mere symbols 
such that 
(3') ai'^, ci^a.2, aiQ:2~, 0^2 

are given the interpretations (3), while any linear combination 
of these products, as 2ai^ — la2^, is interpreted to be the cor- 
responding combination of the a's, as 2aQ — la2,. But no inter- 
pretation is given to a polynomial in ai, 012, any one of whose 
terms is a product of more than three factors a, or fewer than 
three factors a. Thus the first relation (4) does not now follow 
from (3), since the expression 0:1^0:2" (formerly equal to both 

* Due to Aronhold and Clebsch, but equivalent to the more complicated 
hyperdeterminants of Cajdey. 

63 



64 ALGEBRAIC INVARIANTS 

Gofl2 and Gi^) is now excluded from consideration; likewise 
for ai^ao^ and the second relation (4). 

In brief, the general binary cubic (1) may be represented 
in the symbolic form (2) since the products (3') of the symbols 
ai, 02 are in effect independent quantities, in so far as we 
permit the use only of linear combinations of these products. 

But we shall of course have need of other than linear 
functions of oo, • . .,03- To be able to express them sym- 
bolically, we represent / not merely by (2), but also in the 
symbolic forms 

(5) (/3iXi+/32^2)^, (71^1+72:^2)^, . . ., 
so that 

(6) /3i3 = ao, |8i2/32 = ai, Pip2^ = a2, /32^ = a3; 71^ = ^0, .... 

Thus aoCf2 is represented by either ai^/3i/32- or jSi^q;iq:2^, while 
neither of them is identical with the representation ara2^i^^2 
of ai^. Hence 

aoa2-ai2 = i(ai3^i/322+|3i3aia22-2ai2a2/3i2^2) 
= iai/3i(ai^2-a2/3i)2. 

We shall verify that this expression is a seminvariant of 
/. If 

Xi^Xi-{-tX2. .r2=A^2, 

then /becomes F = AoXi^-{-. . . , where 

Ao = ao, Ai=ai-\-tao, A2 = a2-\-2tai-\-fiao, 

^3 = a3+3/a2+3/2ai+/3ao. 
Hence, by (3), 

F = {aiXi-{-a 2X2)^, a 2=oc2-\-tai. 

Similarly, the transform of (5i) is 

(iSiXi +^'2X2)3, |8'2 = i82+^i8i. 

Hence we obtain the desired result 

^0^2-^i2 = |ai/3i(ai^'2-«'2/3i)2 



§40] SYMBOLIC NOTATION 65 

40. General Notations. The binary n-ic 

is represented symbolically as a^" = /3x" = . . . , where 
ax = aiXi-\-a2X2, /3j; = /3i.ri +182X2, . • • , 

a2" = a„; /3i" = ao, .... 
A product involving fewer than n or more than n factors ai, 
0:2 is not employed except, of course, as a component of a 
product of n such factors. 

The general binary linear transformation is denoted by 

T: xi = ^iXi + -niX2, X2 = aA^i + 7,2X2, {kri)9^0 

1 
where {^n) = hm— hvi- It is an important principle of com- 
putation, verified for a special case at the end of § 39, that 
T transforms ax" into the »th power of the linear function 

(ai ^1 -\-0c2 h)X\ + (ai 7/1 +a:2r?2)-X'2 =a^Xi +a,X2, 

which is the transform of a^ by T. Further, 

/j\ «£ oir, ^ ai a2 h V\ 

/?€ ^n iSi 182 * ^2 7/2 
where (ajS) =ai/32 — q:2/3i = — (/3a). Thus 

(a^^,-a,^J)" = (^7,)"(a|8)^ 

so that (ai3)" is an invariant of ai"=fij^ of index «. Since 
{^aY represents the same invariant, the invariant is identically 
zero if n is odd. 



= M)(l^), 



EXERCISES 

1. {a^y is the invariant 2(aoa2 — ar) of ccx- = 0x'^- 

2. (a/3)^ is the invariant 2/ of ax' = /Ji* (§ 31). 

3. (ai3)2 ((37)2 (Ya)2 is the invariant 6/ of ax* = /3i<=7i^ (§ 31). 

4. The Jacobian of ax"* and /3x" is 



7" — ! 

";3x""Vi 



7W — 1 
«|3x" ~^/32 



:m«(«/3)ax'"-Vx'^-^ 



66 



ALGEBRAIC INVARIANTS 



-5. The quotient of the Hessian of ax" = 0x" by «'(« — 1)^ equals 



n —2 9 
ax Oil 

|3/ "-/i,/32 



a 102 



&x 



n-2 



010: 



one-half of the sum of which equals § a^" ^/3j;" '^{a0)'^. 
6. 



ai Pi 7i 

"2 ft 72 

az ft yx 



= M 7:t + (^7)«:r + (7«)ft = 0. 



41. Evident Covariants. We obtain a covariant iiC of 

/ = «/^ = /3/ = . . . 

by taking a product of w factors of type ax and X factors of 
type (a(S), such that a occurs in exactly n factors, jS in exactly 
n factors, etc. On the one hand, the product can be inter- 
preted as a polynomial in ao, • • • , a», ^i, ^2- On the other 
hand, the product is a covariant of index X of /, since, by 
(1), §40, 

{ABy{Acy{Bcy . . . a^b'^c^ . . . 

= i^rjy{a^y{ayy{0yy . . . ax«i8.^7/ • • • , 

if \ = r-\-s-\-t-\-. . . and 

Ax = AiXi-{-A2X2, Ai=a^, ^2=«„ {AB)=AiB2-A2Bi, 

etc. The total degree of the right member in the as, /3's, . . . 
is 2\-\-o: = nd, if d is the number of distinct pairs of symbols 
Oil, a2; /3i, /32; . • .in the product. Evidently d is the degree 
of A' in ao, ai, • • • , and co is its order in xi, X2. 

Any linear combination of such products with the same 
CO and X, and hence same d, is a covariant of order co, index 
X and degree d of /. 

EXERCISES 

1. {a0){ay)ax^l3x'yx*and {ai3y{ay)ax'tix'y'x are CO variants ofax^=ft^= 7x^ 

2. {al5)W "'■ fix"" "'■ is a covariant of ax'\ ft^. 

3. Iim = n, ft"=aa;"^ and ris odd, the last covariant is identically zero, 

4. aoXi^+2aiXiX-+aoXo^ and boXi^+2biXiX2+b2X2^ have the invariant 



§42] SYMBOLIC NOTATION 67 

COVARIANTS AS FUNCTIONS OF TwO SYMBOLIC TypES, §§ 42-45 

42. Any Co variant is a Polynomial in the ax, (a^). This 
fundamental theorem, due to Clebsch, justifies the symbolic 
notation. It shows that any covariant can be expressed in 
a simple notation which reveals at sight the covariant property. 

While a similar result was accomplished by expressing 
covariants in terms of the roots (§36), manipulations with 
symmetric functions of the roots are usually far more complex 
than those with our symbolic expressions. 

The nature of the proof will be clearer if first made for 
a special case. The binary quadratic ax^ has the invariant 

K = aoa2—ai~ 
of index 2. Under transformation T of § 40, ai^ becomes 

(a{Xi+a,A'2)^=/loA'r + . . . , Ao=a(^, Ai=a^a^, A2=a^^. 
Hence AoAo—Ar equals 

We operate on each member twice with 

(1) v = -^- ^ 

9^i9'72 9^29771' 

and prove that we get Q{a0)^ = l2K, so that K is expressed 
in the desired symbolic form. We have 



(^^) = ^l'72-^2 



VI, 



:^(^r?)2 = 2(^77)a, -r^(^vy = 2i^v)+2n^^, 

dV2 0K10V2 

OVl dK20r]l 

V{^vy = Q{^v), F2(|r,)2 = 12, 

since F(^r?) =2, by inspection. Next 
(2) Va^^r, = V{ai ^1 +a2 ^2) (Pi Vi +132^72) =«il82 -a2^i = (oc0). 



68 ALGEBRAIC INVARIANTS 

Hence 

= /3ea,(a^)+aj^,(i3«), 

The difference of the expressions involving V^ is 6(q:/3)2. Hence 
if (1) operates twice on the equation preceding it, the result is 

43. Lemma. F'^(^t7)" = (w + 1)(w!)2. 

We have proved this for w = 1 and n = 2. li n'L2, 

9?i9'72 
Similarly, or by interchanging subscripts 1 and 2, we get 



Subtracting, we get 

F(^T?)" = !2w+w(w-l)}(^r7)"-i=w(;^ + l)(^r7)"-i. 
It follows by induction that, if r is a positive integer, 

F'-(^7?)"=(w + l)lw(fi-l) . . . (w-r+2)|2(w-r+l)(^7J)"-^ 
The case r = n yields the Lemma. 

44. Lemma. // the operator V is applied r times to a product 
of k factors of the type aj and I factors of the type /3„ there results 
a sum of terms each containing k—r factors a^, l—r factors )3„ 
and r factors {a0) . 

The Lemma is a generalization of (2), § 42. To prove 
it, set 



§44] SYMBOLIC NOTATION 69 

Then 



(s) a (<)' 






a?29^i . = 1^ = 1 aj(^^/3,^'^' 

Subtracting, we get 

Hence the lemma is true when r = l. It now follows at once 
by induction that 

(1) VAB 

A B 



= S2(a(si)|3('i)) . . . {a(sr)^{tr)). 



«l 



(si) . . . ajCsO |3,('i) . . . |S,('0' 



where the first summation extends over all of the 
hik — l) . . . (^' — r + 1) permutations ^i, . . . , ^r of 1, . . . , ^ 
taken r at a time, and the second summation extends over 
all of the /(/ — I) . . . (/ — r + 1) permutations /i, . . . , Ir of 
1, . . . , / taken ;' at a time. 

Corollary. The terms of (1) coincide in sets of rl and 
the number of formally distinct terms is 

II 1 /k\/l\ , 
rl. 



{k-r)\ {l-r)\ rl \r/\r 

For, we obtain the same product of determinantal factors 
if we rearrange si, . . . , Sr and make the same rearrangement 
of /i, . . . , tr. 

45. Proof of the Fundamental Theorem in § 42. Let K be 

a homogeneous covariant of order cu and index X of the binary 
form / in § 40. By § 40, the general linear transformation 
replaces /=a:x" by 



(1) ^t = ai"-V (^ = 0,1,. . .,n). 



k=o\k 
Hence 



70 ALGEBRAIC INVARIANTS 

By the covariance of K, 
(2) K{Aq, . . . ,An\ Xi, X2) = {h)^K{aQ, . . . , a„; :ki,:^2). 
By (1) the left member equals 

»'=0 

in which the inner summation extends over various products 
AB, where ^ is a product of a constant and factors of type 
a|, and 5 is a product of a constant and factors of type a,. 
Let Xi=y2, and X2^—yi. Then, by solving the equations 
of T, § 40, 

Xi=yJ{^v), X2=-yi/{^ri). 

Hence the equation (2) becomes 

i 2{-iyABy„'^-'yii={^vy+''K. 

t=0 

Since the right member is of degree X + co in ^i, ^2, and of 
degree \-\-co in 771, 772, we infer that each term of the left mem- 
ber involves exactly X + co factors with subscript ^ and X+w 
factors with subscript rj. 

Operate with F^"*"" on each member. By § 43, the right 
member becomes cK, where c is a numerical constant 5^0. 
By § 44, the left member becomes a sum of products each of 
X+co determinantal factors of which co are of t^'pe {ay) = ax, 
and hence X of type {a0). The last is true also by the definiton 
of the index X of K. Hence K equals a polynomial in the 
symbols of the types ax, (a0). 

To extend the proof to covariants of several binary forms 
a/, yx^, . . . , we employ, in addition to (l),Ct = 7j*"~*7,*, • • • 
and read a^, y^, . . . for a^ in the above proof. 

FiNITENESS OF A FUNDAMENTAL SYSTEM OF COVARIANTS, 

§§ 46-51 

46. Remarks on the Problem. It was shown in §§28-31 
that a binary form / of order <5 has a finite fundamental 
system of rational integral covariants A'l, . . . , Ks, such 
therefore that any rational integral covariant of / is a poly- 



§ 47] FINITENESS OF COVARIANTS ; 71 

nomial in Ki, . . . , Kg with numerical coefl&cients. We shall 
now prove a like theorem for the covariants of any system 
of binary forms of any orders. The first proof was that by 
Gordan; it was based upon the symbolic notation and gave 
the means of actually constructing a fundamental system. 
Cayley had earlier come to the conclusion that the fundamental 
system for a binary quintic is infinite, after making a false 
assumption on the independence of the syzygies between the 
covariants. The proof reproduced here is one of those by 
Hilbert; it is merely an existence proof, giving no clue as to 
the actual covariants in a fundamental system. 

47. Reduction of the Problem on Covariants to one on In- 
variants. We shall prove that the set of all covariants of the 
binary forms /i, ...,/* is identical with the set of forms 
derived from the invariants 7 of /i, . . . , ft and l^xy' — x'y 
by replacing .r' by x and y' by y in each /. It is here assumed 
(§ 15) that 7 is homogeneous in the coefficients of / and that 
the covariants are homogeneous in the variables. 

Let the coefiicients of the /'s he a, b, ... , arranged in 
any sequence. Let A, B, . . .be the corresponding coefficients 
of the forms obtained by applying the transformation in § 5. 
The latter replaces / by W-^'t], where 

r,'=ay'-yx', ^' = bx'-^y'. 

Solving these, we get 

Ax'=a^'+^r?', ^y' = yt'J^bn'. 

Let I{a, b, . . . ; x', y') be an invariant of / and the /'s. 
Then 

I(A,B,. . .; ^\r,')=AH(a,b,. . .; x',y'). 

Since 7 is homogeneous, of order w, in x', y', the right member 
equals 

A^-"7((Z, b, . . . ; Ax', Ay')- 

Hence we have the identity in ^', rj' : 

liA, B,. . .; ^', r?0^A^-<::7(a, b, . . . ■ a^'+/3r?', y^'+W). 



72 ALGEBRAIC INVARIANTS 

Thus we may remove the accents on ^', t] . Then, by our 
transformation, 

I{A,B,. . . ; ^,v)=A^-"I{a,b,. . .; x,y). 

Hence I {a, b, . . . ; x, y) is a. covariant of /i, . . . , ft oi order 
o) and index X — co. * 

The argument can be reversed. Note that the sum of the 
order and the index of a covariant is its weight (§ 22) and hence 
is not negative. 

Corollary. A covariant of the binary form / has the 
annhilators in § 23. 

For, an invariant of / and xy' — x'y has the annihilators 

fi-y--, o-x'^, 

ax" ay' 

48. Hilbert's Theorem. Any set S of forms in xi, . . . , Xa 
contains a finite number of forms Fi, . . . , F^ such that any form 
F of the set can be expressed as F=fiFi-\-. . .4-/t^t, where 
fx, . . . , fk drc forms in xi, . . . , Xn, but not necessarily in 
the set S. 

For w = l, 5 is composed of certain forms Cix^\ C2X^\ .... 
Let Cs be the least of the e's, and set Fi=CsX^K Then each 
form in 5 is the product of Fi by a factor of the form cx«, e ^ 0. 
Thus the theorem holds when n = l. 

To proceed by induction, let the theorem hold for every 
set of forms in n — \ variables. To prove it for the system 
S, we may assume, without real loss of generality,* that S 
contains a form Fq of total order r in which the coefficient 
of xj is not zero. Let F be any form of the set 5. By division 
we have F = FqP-\-R, where i? is a form whose order in Xn 

* Let F be a form in 5 not identically zero and let the linear transformation 

Xi = Cii3'i+Cj2>'2+. . .+Cinyn (/=L. . . , «) 

replace F(:x:i, . . . ,Xn) by K{yi, . . . , jn)- In the latter the coefficient of the term 
involving only >'« is obtained from F by setting Xi = cin and hence is F{cin., con, ■ ■ • , 
Cnn), which is not zero for suitably chosen c's (Weber's Algebra, vol. T, p. 457; 
second edition, p. 147). But our theorem will be true for 5 if proved true for 
the set of forms K. 



§48] FINITENESS OF COVARIANTS 73 

is <r. In i? we segregate the terms whose order in x» is exactly 
r — 1, and have 

where M is a form in xi, . . . , a-„_i, while iV is a form in 
iCi, . . . , Xn whose order in Xn is :^ /- — 2. Each F uniquely deter- 
mines an M. 

For the defmite set of forms M in w — 1 variables the theorem 
is true by hypothesis. Hence there exists a fmite number 
of the M's, say M\, . . . , Mi (derived from Fi, . . . , F^, 
such that any M can be expressed as 

where the/'s are forms in xi, . . . , x„_i. Then 

t = 1 

F = FoP'-{- ^fiF^+R', P'^P-Zf^P^, R'^N-Xf,Ni. 

1=1 

Each exponent of Xn in R' is ^r — 2. We segregate its terms 
in which this exponent is exactly r — 2 and have 

F = FoP'-i- i JiF.-hM'xZ-'^+N', 
«=i 

where M' is a form in xi, . . . , Xn-i, and N' a form in 
Xi, . . . , x„ whose order in x„ is ^ r— 3. 

The theorem is apphcable to the set of forms M\ so that 
each is a linear combination of M\, . . . , M'm, corresponding 
to i^'^+i, . . . , Fi+jn, say. As before, F differs from a linear 
combination of Fq, . . . , Fi+mhy 

M"x/-^-\-N", 

where M" is a form in xi, . . . , Xn-i and TV" is a form whose 
order in Xn is ^ r — 4. Proceeding in this manner, we see that 
F differs from a linear combination of Fq, . . . , Ft by a form 
R in xi, . . . , x„^i. One more step leads to the theorem. 

49. Finiteness of a Fundamental System of Invariants. Con- 
sider the set of all invariants of the binary forms /i, . . . , fa, 



74 ALGEBRAIC INVARIANTS 

homogeneous in the coefi&cients of each form separately. By 
the preceding theorem, there is a finite number of these invariants 
/i, . . . , Im in terms of which any one of the invariants / is 
expressible linearly: 

(1) / = £i/i + . . .-\-EmIn., 

where Ej is not necessarily an invariant, but is a polynomial 
homogeneous in the coefficients of each fi separately. 

Let ai, a2, ... be the coefficients in any order of/i, . . . ,/d. 
Let A I, A2, . . . be the coefficients in the same order of the 
forms obtained from them by applying a linear transformation 
of determinant (l??). We may write 

I{A) = {^r,)^I{a), Ij{A) = avrjTj{a), EAA)=Gj, 

where Gj is a function of the a's, |'s, 77's. From the identity 
(1) in the a's, we obtain an identity by replacing the a's by the 
A's. Hence 

m 

ihYl^ ^Gji^vYJlj, 

in which the arguments of the /'s are a's. Thus Gj is of order 
X — X; in ^1, I2 and of order X — X^ in rji, 772. Operate on each 
member by V^. By § 43, the left member becomes 

(X + 1)(X!)2/. 

By the formula to be proved in § 50, the right member 
becomes 

:^IACo(^r,)'^-^Gj-^C,{h)^^-^+'VGj-h. . .+a{^vY^V'Gj\, 

i = i 

where the C's are numerical constants. Since Gj is of order 
v = X — X^ ^ in ^1, ^2 and of order p in 771, 772, 

V+'Gj^O, V^+-Gj = 0, ..., V^Gj = 0. 

Also Co, Ci, . . . , Cy-i are zero since they multiply powers 
of (^rj) whose exponents —v, —v-\-l, . . ., X^ — X+j^— 1= — 1 
are negative. Hence 

m 

(X+l)(X!)2/= ZljC^VGj. 



§50] FINITENESS OF COVARIANTS 75 

The torm obtained from/t = Q;2" by our linear transformation 
has the coefficients (1), § 45. The polynomial G] in these 
coefficients is therefore a sum of terms each a product of a 
constant by v factors of type a^ and v factors of type or,. 
Hence, by § 44, VCj is a polynomial in the determinantal 
factors (a/S) and is consequently an invariant of the forms 
fi. Thus 

m 

1= S//0, 
where I'j is an invariant. Then, by (l), 

TO m 

k=\ i,k=\ 

By repeating the former process on this /, we get 

m 
j,k=\ 

where the /" are invariants of the forms ft. Since there is 
a reduction of degree at each step, we ultimately obtain an 
expression for / as a polynomial m Ii, . . . , In with numerical 
coefficients. 

50. Lemma. If D= ^ir]2— ^2Vi, cind P is homogeneous {of 
order X) in |i, ^2, and homogeneous {of order n) in t/i, 772, then 

m 

(1) F"»Z)"P= 2 GD" -'"+'■ FP, 

r=0 

where Co, . . . , Cm are constants. 
First, we have 

FZ)P = P + r/2— +^1— +/>- ^^^ 



'dri2 dh dhdm 

-{-P-h^-ml-+D:^)=-2+X+^)P+DVP, 

\ dh 9171 9^29771/ 

by Euler's theorem for homogeneous functions (§24). If P 
is replaced by D"-^P, so that X and m are increased by n — 1, we 
get 

VD"P = {\-\-fx-\-2n)D''-^P-\-DVD''-'^P. 



76 ALGEBRAIC INVARIANTS 

Using this as a recursion formula, we get 

which reduces to the result in §43 if P = l, whence X = /i = 0. 
Hence (1) holds when m = l. To proceed by induction from 
m to w + 1, apply V to (1). Thus 

m 
r=0 

In the result for VD"'P, replace n by n—m-\-r and P by 
V^P, and therefore diminish X and m by r. We get 

Yf£)n-m+rYrp\ _^ £)n -m+r -lyrp \£)n—m+rYr + lp 

where 

tr={n—m-\-r){\-\-ij. — r-{-n—m-\-l). 

Hence, changing r+l to r in the second summand, we get 

m+l 
ym+lj)np^ V (Crtr + Cr-l)D''-'^+'-^V'P, 
r=0 

with C„,+i = 0, C_i = 0. Thus (1) is true for every m. 

51. Finiteness of Syzygies. Let /i, . . . , 7^ be a funda- 
mental system of invariants of the binary forms /i, . . . , fa- 
Let S{zi, . . . , Zm) he a. polynomial with numerical coefficients 
such that S(Ii, . . . , /w),Vhen expressed as a function of the 
coeflficients c of the /'s, is identically zero in the c's. Then 
5(7) = is a syzygy between the invariants. 

By means of a new variable z^+i, construct the homogeneous 
form 5'(zi, . . ., z^+i) corresponding to S. By §48, the 
forms S' are expressible linearly in terms of a finite number 
S\, . . . , yt of them. Take 2^+1 = L Thus 

(1) 5 = Ci5i + . . .+CA 

where Ci, . . . , Ct are polynomials in zi, . . . , z„,. Take 
zi =Ii, . . . , Zm = Im- Hence there is a finite number of syzygies 
Si=0, . . ., St = 0, such that any syzygy 5 = implies a 
relation (l) in which Ci, . . . , C* are invariants. In particular, 
every syzygy is a consequence of 5i = 0, . . . , 5t = 0. 



§ 52] TRANSVECTANTS 77 

52. Trans vectants. Any two binary forms 

have the covariant 

(1) (/, 0)'- = (a^)'-«/-'-/3i-'-, 

called the rth. transvectant (Ueberschiebung) of / and <^, and 
due to Caj'ley. It is their product if r = 0, their Jacobian 
if r = l, and their Hessian \i J=<t> and r = 2, provided numerical 
factors are ignored (Exs. 4, 5, § 40). 

It may be obtained by differentiation and without the use 
of the symbolic notation. In fact, a special case of (l), § 44, is 

so that if/ is of order k and 4> of order /, 

(2) im, 0(^))^ = fc^' ^^[F'-/(^)c/>fr,)l, = ,. 



x\ftcr/(ti, ^2)-4>(m, m) is operated on by V, we set 771 = ^1, 
For example, let /(^)=a^a^, </.($) = 7^\ P^a^S^y '. Then 



d^idvi ^ * 9^29771 * * 

The difference is VP. Taking r]i= ^i, 772= I2, we get 

The numerical factor in (2) is here I/6. Hence 

(3) ^<^j y,')' = h{0yh^yf+K'xy)0^y^'. 

In general, consider the two forms 
Then by (l), § 44, and the Corollary, and by (2), 

where the summation extends over all the combinations of the 



78 ALGEBRAIC INVARIANTS 

as r at a time, and over all the permutations of the /S's r at 
a time. Thus the number of terms in the sum is the reciprocal 
of the factor preceding S. 

If the a's are identified and also the /3's, (4) becomes (1). li k = 2, 
l=S, r=l, we have one-sixth of a sum of six terms; then if the /3's are 
identified we have two sets of three equal terms and obtain (3). 

Since F is a differential operator, (2) gives 
(5) (2c/,, i:kj<f>jy = ^^cMft, 4>,)r. 

Apolarity; Rational Curves, §§ 53-57 

53. Binary Forms Apolar to a Given Form. Two binary 
quadratic forms are called apolar if their lineo-linear invariant 
is zero; then they are harmonic (Ex. 3, § 11). In general, 
the binary forms 

i=Q\l/ i=o\t/ 

of the same order, are called apolar if 

(1) {a^y'^=lji-i)'(jyA.-i=o. 

In particular, / is apolar to itself if n is odd (Ex. 4, § 38). 

Let the actual linear factors of <^ be /3x^^\ • • • , /3j'^"^ By 
(1), (4), § 52, 

(a^)" = (a/, /3,(i> . . . /3,("))" = (a|3(i)) . . . (a/3("^). 
But ^x^"^ vanishes if xi and X2 equal respectively 

Thus 

Hence if vanishes for xi=yi''''\ X2=y2^''^ (r = l, . . . , n), 
it is apolar to f if and only if 

Thus / is apolar to an actual wth power (y2A;i— ^1^2)" if 
and only if q:j,'* = 0, i.e., if yi, y2 is a pair of values for which 
/=0. 



§ 53] APOLARITY 79 

If no two of the actual linear factors /< of / are propor- 
tional, / is apolar to n actual «th powers //' and these are readily 
seen to be linearly independent. Then their linear combinations 
give all the forms apolar to/. For, if/ is apolar to <^i, . . . , 0n, 
it is apolar to /^i<^i + . . •-\-kn4>n^ where ^i, , . . , ^a are con- 
stants, since, by (5), § 52, 

(/, y^i0i+. . .+K^nY = k,{J,<j>iY+. . .-\-knU,4>nY = o. 

Moreover,/ is not apolar to n-\-\ linearly independent forms 

01, 02, • • . , 0n+l- 

For, if so, we have n-\-l equations like (1), in which the deter- 
minant of the coefficients of ao, • • • , an is therefore zero. 
But this implies a linear relation between the 0's. /// is the 
product of n distinct linear factors U, a form can be repre- 
sented as a linear combination of /i", . . . , In^ if afid only if 
is apolar to f. In particular, if r and 5 are the distinct roots 
of f^ax'-\-2bx-\-c = 0, the only quadratics harmonic to / are 
g{x-rY+}i{x-sY. 

In case h, . . . , Ir are identical, while li^lt{i>r), we may 
replace /i", . . . , /r" in the above discussion by /i", /i"~^X, . . . , 
l^n-r+iy-i^ whcrc X is any linear function of Xi and x-y which 
is hnearly independent of h. In fact, after a linear trans- 
formation of variables, we may set li=X2, \ = Xi. Then the 
above r forms have the factor 0:2""''"^^ and hence are of type 
with bi = 0{i ^ n—r). Also, / now has the factor x-z^, so 
that ai = 0{i<r). Hence every term of (l) is zero. 

For example, /=.Vi-.r-..(.Vi — .Vi)- is apolar to 

Xi^, Xi^Vi] .r2»; (xi — x-2)\ (.fi — .^2)^^:1, 
which give five linearly independent quintics. 

In general, when there are multiple factors of /, the n 
forms apolar to / obtained above can be proved to be linearly 
independent. This fact is not presupposed in what follows. 

54. Binary Forms Apolar to Several Given Forms. From 
the list of the given forms we may drop any one linearly de- 



80 ALGEBRAIC INVARIANTS 

pendent on the others, since a form apolar to several forms is 
apolar to any Hnear combination of them. In the resulting 
linearly independent forms 

the g-rowed determinants in the rectangular array of the 
coefficients are not all zero. For, if so, there are solutions 
^1, . . . T kg, not all zero, of 

kiaii-\-k2aio-\- . . .-\-kgaig = {i = 0,l,. . . ,n), 

which would give, contrary to hypothesis, the identity 

If io^i" + . . . is apolar to each/r, then 



2(-l)Y".)aJ;„_,- = (r = l,. . . , g). 



These determine g of the b's as linear functions of the remaining 
5's, which are arbitrary. Hence there are exactly n-\-l—g 
linearly independent forms apolar to each of the g given 
linearly independent forms. 

In particular, apart from a constant factor, there is a 
single form apolar to each of n given linearly independent 
forms of order n. 

Consider three binary cubic forms 

/i =aJ^=aoXi^+3aixrx2-}-^a2XiX2^-\-a3X2^, 

J2 = /Sx^ = h(iXi^ -\-2,hxXi^X2-\-^h2XiX2'^ +h2,X2^ , 
Jz=-'yx^^CQXi^-\-ZCiXi~X2+2>C2XiX2^-{-C2X2^. 

Each is apolar to the cubic form 

= (a/3)(a7)(i37)«:ri3^Tx. 
For, by (4), § 52, and the removal of a constant factor by (5), 

(<^, 5.-^)3 = («i3)(a7)(/3T)(«5)(/35)(75), 
which is changed in sign if 6 is interchanged with a, /3, or 7, 



§54] 



APOLARITY 



81 



and hence is zero if bj^ is one of the fi. Hence each /< is apolar 
to 4). Now 

ai" a\a.2 of^ 

71^ 7172 72" 

In fact, the determinant vanishes if (ajS) = as may be seen 
by setting ^i=cai, ^2=ca2. Moreover, the two members are 
of total degree six and the diagonal term of the determinant 
equals the product of the first terms 0:1182, etc., on the left. 

Since ai^ax=ai^xi-i-ai^a2X2=aoXi-{-aiX2, etc., we find, by 
multiplying the members of the last equation by ax/3x7z, 

aoXi-j-aiX2 ai:ri+a2^2 02^1+03X2 

boXi-\-biX2 biXi-{-b2X2 b2Xi-\-b3X2 
CoXi-\-CiX2 ^1:^1+^2^:2 C2X1+C3X2 



where 



= [012]xl3 + [013]A-l%2 + [023]xlX22 + [123]:^;2^ 



at Qj at 
[ijk] = 1 bi bj bt 

Ci Cj Ct 

If (p is identically zero, the four three-rowed determinants 
in the rectangular array of the coefficients of /i, /2, /a are all 
zero, and the/'s are linearly dependent. 

Apart from a constant factor, is the unique form apolcr 
to three linearly independent cubic forms fi, f 2, fz- 

The extension to n binary w-ics is readily made. 

55. Rational Plane Cubic Curves. The homogeneous coor- 
dinates ^, rj, f of a point on such a curve are cubic functions 
of a parameter /. We may take / = a:i/.T2 and write 

Ps=/l, PV^f2, P^^fd, 

where p is a factor of proportionality and the /'s are the cubic 
forms in § 54. 

We may assume that the/'s are linearly independent, since 
otherwise all of the points (^, tj, f) would He on a straight 
line. 



82 ALGEBRAIC INVARIANTS 

There is a unique cubic form ^ apolar to /i, /2, /a (§ 54). 
This cubic form, denoted by (p = (p/, is fundamental in the 
theory of the cubic curve. 

Three points determined by the pairs of parameters xi, X2; 
yi, y2', and z\, so, are collinear if and only if 

(1) 4>x4>Az = ^- 

For, if the three points he on the straight Hne 

(2) /t + ;«r7 + «r = 0, 

the three pairs of parameters are pairs of values for which 

(3) C{xi, x-f) ^lfi-^mf2-{-nf3 = 0. 

Since C is apolar to (p, (1) follows from the first italicized theorem 
in § 53. Conversely, (1) implies that the cubic C which van- 
ishes for the three pairs of parameters is apolar to ^ and hence 
(§ 53) is a linear combination of /i, /2, /s, say (3); the corre- 
sponding three points lie on the straight line (2). 

Since (2) meets the curve in three points the ratios xi/x-z 
of whose parameters are the roots of (3), the curve is of the 
third order. 

We restrict attention to the case in which the actual linear 
factors ax, ^x, ix of 4> are distinct. Since any cubic apolar 
to </> is a linear combination of their cubes (§ 53), 

/i = rna.3+Ct2/3/+c,37/ (^ = 1, 2, 3). 

Since the determinant \ dj \ is not zero, suitable linear com- 
binations of the/'s give a/, /3x^, jx^. Hence by a linear trans- 
formation on ^, 7], f (i. e., by choice of a new triangle of ref- 
erence), we may take * 

The line ^ = is an inflexion tangent, likewise 7j = and 
f = 0. In addition to the resulting three inflexion points, 
there are no others. For, at an inflexion point three consecutive 
points are collinear, so that (l) gives (^ = <^/ = 0. In the present 

* We now have the formulas in the second part of § 54, where now ax^ is the 
actual, not a symbolic, expression of /i, etc. 



§56] RATIONAL PLANE CUBIC CURVE 83 

case there are therefore exactly three inflexion points and they 
are coHinear. 

56. Any Rational Plane Cubic Curve has a Double Point. 

Let Px denote the point (^, r/, i") determined by the pair of 
parameters xi, xo. If the ratios x\/x2 and yi/y2 are distinct 
and yet Pi coincides with Py, then Px is a double point. For, 
any straight line (2), § 55, through Px meets the curve in 
only the three points whose pairs of parameters satisfy the 
cubic equation (3), and since two of these pairs give the same 
point Px. the line meets the curve in a single further point. 
Hence there is a double point Px = Py if and only if there are 
two distinct ratios xi/xo and yi/>'2 such that (l) holds identically 
in Si, 22- 

Let Q be the quadratic form which vanishes for the pairs 
of parameters xi, X2 and yi, yo giving a double point. By (1), 
and the first theorem in § 53, Q is apolar to <f)x~(f>z for Si, Z2 
arbitrary. Write (f)'i^ as a symbolic notation for <f>, alter- 
native to ^/. Applying the argument made in § 54 for three 
cubics to two quadratics, we see that the unique quadratic 
(apart from a constant factor) which is apolar to both <^^-0, 
and (t>'x'^4>'u> is their Jacobian 

/= (<^0 )<t>x4> X* 020 W 

Since and 0' are equivalent symbols, their interchange must 
leave / unaltered. Hence 

-^ = 2(00 )0x0 x{(t>z4>'w — 4>'z4>tc\. 

The quantity in brackets equals {<i><i>'){zw) by (l), § 40. Dis- 
carding the constant factor \{zw), we may take 

as the desired quadratic form. This is the Hessian of 0. 
Conversely, the pairs of values for which Q vanishes are the 
pairs of parameters of the unique double point of the curve. 

57. Rational Space Quartic Curve. Such a curve is given 
by 



84 ALGEBRAIC INVARIANTS 

where the four binary quartics are linearly independent. By 
§ 54, there is a unique quartic apolar to each of the four. 
As in § 55, four points Px, Py, Pz, Pw on the curve are coplanar 
if and only if 

Thus = gives the four points at which the osculating plane 
meets the curve in four consecutive points. It may be shown 
that the values i"/*\ X2^'^^ for which the Hessian of </> vanishes 
give the four points i'j:^') on the curve the tangents at which 
meet the curve again. 

Fundamental Systems of Covariants of Binary Forms 

§§ 58-63 

58. Linear Forms. A linear form otx is its own symbolic 
representation. If ax = 0x, then (q:/3)=0. Hence the only 
covariants of ax are products of its powers by constants. A 
fundamental system of covariants of n linear forms is evidently 
given by the forms and the ln(n — l) invariants of type (a/3), 
where ax and 0x are two of the forms. 

59. Quadratic Form. A covariant K of a single quadratic 
may have no factor of type (a/S) and then it is 

ax Px Ifx • . • J , 

or may have the factor (a/S) and hence the further factor {a0), 
{ay){p8), {ay)^x, or axl3x, including the possibility 5 = 7. In 
the first case, K^{a^)^Ki, where A'l is a covariant to which 
the same argument may be applied. Now {a'y)^ay if ^1=72, 
y2=—yx. Hence in the last three cases, K has a factor of 
the type 

e = {a^)ay^^, 

where ay is either ax or a new mode of writing (0:7), and similarly 
jSj is either ^x or a new mode of writing (/35). 

Interchanging the equivalent symbols a and 8, we get 

d - i^a)^,a, = K«/3) (a,/3, - /3,«.) = § (a^)2(y3), 



§eO] FUNDAMENTAL SYSTEM OF COVARIANTS 85 

by (1), § 40. We are thus led to the first case. Hence the 
fundamental system of covariants of / is composed of / and 
its discriminant. 

EXERCISES 

1. The fundamental system for f=ax^ = bx- and l=ax = 0x is /, /, {ab)*, 
{aocy, {aa)ax. 

2. The fundamental system for f=ax^ = bx- and <t> = ax^ = Px^ is/, <t>, 
{aby, (a^)^, (oa)-, {aa)axax. Hint: 

{act) {a0)aaliy = {aa) '-fiy^z-\{a0) ^GyOz, 

as proved by multiplying together the identities (Ex. 6, § 40) 

{a^)ay = {ap)ay-{aa)0^, (a/3)(Zz=(d/3)as-(aa)/5,, 

and noting that a and /3 are equivalent symbols. 

60. Theorems on Trans vectants. In the expression (4), 
§ 52, for a transvectant, each summand taken without the 
prefixed numerical factor is called a term of the transvectant. 
In the first transvectant (3), § 52, the difference of the two 
terms is 

by Ex. 6, § 40, and is the negative of the 0th transvectant 
(viz., product) of (a/3) and y^^. The act of remo\dng a factor 
a^ and a factor jS^ from a product and multiplying by the 
factor (q:/3) is called a convolution (Faltung). We have therefore 
an illustration of the following 

Lemma. The difference between any two terms of a trans- 
vectant equals a sum of terms each a term of a lower transvectant 
of forms obtained by convolution* from the two given forms. 

Consider the rth transvectant of 

where P and Q are products of determinantal factors. Then 
PQ is a factor of each term of the transvectant. Any two 
terms T and T differ only as to the arrangements of the as 
and the /3's. Hence T' can be derived from 7 by a permuta- 

* Including the case of no convolution, as 7^' from itself, in the above example. 



86 ALGEBRAIC INVARIANTS 

tion on the as and one on the /S's, and hence by successive 
interchanges of two a's and successive interchanges of two 
jS's. Any such interchange is said to replace a term by an 
adjacent term. For example, the two terms of (3), § 52, 
are adjacent, each being derived from the other by the inter- 
change of a with /3. Between T and T' we may therefore 
insert terms Ti, . . . , Tn such that any term of the series 
T, Ti, T2, ■ ■ ' , Tn, T' is adjacent to the one on either side 
of it. Since 

r-r=(r-ri)+(ri-r2)+. . .+(r„_i-r„)+rr„-r), 

it suffices to prove the lemma for adjacent terms. 

The interchange of two a's or two /3's affects just two factors 
of a term of (4), § 52. The types of adjacent terms are * 

where /3' and /3" were interchanged. The difference of the 
last two terms is seen to equal C(jS"^')a\ by the usual identity. 
The latter is evidently a term of the (r — l)th transvectant 
of / and {l3"^')4>/{^'\^\], which is obtained from <f) by one 
convolution. 

The difference of the first two adjacent terms equals 
C(aW')(fi'^"), since 

a la I p i p 1 



{a'a")(p'n - ia'^'){a"n + icc'fi"){a"^') ^\ 



I n n> o" 

a 2OL 2P2P 2 

' // Qt o" 
a \a i p 1 p 1 

a 20i 2 P 2 p 2 



= 0, 



as shown by Laplace's development. The same relation 
follows also from the identity just used by taking |i = — a"2, 
^2= a" I. The resulting difference is a term of the (r-2)th 
transvectant of 

a ^a ^ p iP ^ 

which are derived from / and by a convolution. 

*A pair C(a'0')a"^, C(a"li')a'^, obtained by interchanging a and a", is 
essentially of the second type. 



§G0] FUNDAMENTAL SYSTEM OF COVARIANTS 87 

The LcMnma leads to a more important result. By tlie 
proof leading to (4), § 52. the coefficient of each term of a 
transvectant is 1/N, if .Y is the number of terms. Just as 
S = hiTi-\-T-2) implies S-Ti=h(T2-Ti), so 



impUes 



S = ^{Ti + . . . + r.v) 






Hence ilic clijjcrciicc bclwccii a traiisvccia>i! a)iJ any one of its 
terms equal a sum of lernis eaeJi a term of a lower iransveeiant 
of forms obtained by eonvolntion from the two givefi forms. 

Each term of a lower transvectant may be expressed, by 
the same theorem, as the sum of that transvectant and terms 
of still lower transvectants, etc. Finally, when we reach a 
0th transvectant, i.e., the product of the two forms, the only 
term is that product. Hence we have the fundamental 

Theorem. The differ enee between any transveetant and 
any one of its terms is a linear function of lower Iransvcclanls 
of forms obtained by convolution from the two given forms. 

For example, from (3), §52, and the result preceding the 
Lemma, we have 

and {ajS) is dcri\oil from (Vcp\ by one convolution. 

61. Irreducible Covariants of Degree m Found by Induction. 
Let 

/ = a/ = /3V' = . . .=X/ 

be the binary ;;-ic whose fundamental s}'stem of covariants 
is desired. Since a term with the factor (a/3) is of degree at 
least two in the coefficients of /, the only covariants of degree 
unity are kf, where k is a numerical constant. We shall say 
that / is the only irreducible covariant of degree unity, and 
that/, A'l, .... /v, form a complete set of irreducible covariants 
of degrees <m if every covariant of degree <;;/ is a poly- 



88 ALGEBRAIC INVARIANTS 

nomial in f, ..., Ks with numerical coefficients. Given the 
latter, we seek the irreducible co variants of degree m. 

A covariant of degree m is a polynomial in the (a/S) and 
the ax such that each term contains m letters a, ^, y, . . . . 
Let Tm be one of the terms with its numerical factor suppressed. 
Let a, /3, . . . , K, X be the m letters occurring in Tm, so that 

r„ = P(aX)«(/3X)^ . . . {kXYxJ {a-\-b + . . .+k-^l = n), 
where P involves only a, /3, . . . , k. Then 

r^_i=Pa.«/3/ . . . K,fc 
is a covariant of degree m—l. Evidently Tm is a term of 

{Tm-i, X/)^ {r = n-l), 

since it is obtained by r = a-\-b^. . .-{-k convolutions from 
Tm-i^x"^- By the final theorem in § 60, 

Tm={Tm-ljy+^^'cj{frn-ufy, 
j=0 

where the Cj are numerical constants, and each Tj^-i is derived 
from r„i_i by convolutions and hence is a covariant of degree 
m—l. But the covariant of degree m was a linear function 
of the various Tm- Hence every covariant of degree m oj f is 
a linear function of transvectants {Cm-ufY of covariants C^-i 
of order m—l with f. Such a transvectant is zero if k>n, 
in view of the order of/. Moreover, it suffices by (5), §52, 
to employ the C^-i which are products of powers oif,Ki,. . . , 
Kg. Hence the covariants of degree m are linear functions of 
a finite number of transvectants. 

In the examination of these transvectants {Cm-i, fY, we 
first consider those with k = \, then those with k = 2, etc. We 
may discard any {Cm-i,fY for which Cm-i has a factor <t>, of 
order '^k, which is a product of powers of f, Ki, . . . , Ks, and 
of degree <m-l. For, if T is a term of (c^,/)^, and if C^_i=^0, 
then T is obtained by k convolutions of 0/, and qT by the same 
k convolutions of g<^/, not affecting q. Hence qT is a term of 
{q(t>, fY' Hence 

(Cm-i, fY-qT+~^' Cj(Cm-i,fy. 

3=0 



§ G2] FUNDAMENTAL SYSTEM OF CO VARIANTS 80 

But the terms of the last sum have by hypothesis been con- 
sidered previously, while the co variants ^ and T are of degree * 
<m and hence are expressible in terms of/, Ki, . . . , Ks. 

62. Binary Cubic Foim. The only irreducible co variant of 
degree one of 

was shown to be /. The only covariants of degree two are 

(a^)'-a/-%-'^-'- (r = 0, 1, 2,3). 

This vanishes identically if r is odd. If r = 0, we have /-, 
which is reducible. Hence the only irreducible covariant of 
degree two is 

ia0)-a,l3, = (/, f)~ = Hessian H of /. 

To find the irreducible covariants of degree m=S, we 
have Cjn-i = Ii or/-. In the second case, C^-i has the factor 
f of degree <m — l and order 3^^' (since we cannot remove 
by convolution more than three factors from the second function 
f in the transvectant). Hence we may discard Cm-i=f~- It 
remains to consider (H, f)^\ k^l, 2. Now 

(H, /) = (a/3)2(a7)/3^7/ = Jacobian / of // and / 

is irreducible, being of order and degree three and hence not 
a polynomial in / and H. Next, 

{H, jy - (a/3)2(a7) (^7)7x = P{a^h., P = {cc0) {ay) (fiy) . 

Interchanging a with 7, we get P{^y)ai. Interchanging jS 
with 7, we get P(y(x)0j. Hence 

(F,/P = |F|M)7. + (^7K + (7«)/3x|=0. 

The irreducible covariants of degree three or less are therefore 
/, H, J. 

To find those of degree 7;z=4, we have Cm-i=P, fB, J, 

* This is evident for the factory of Cm-i- Since <f) is of degree <w— 1, the 
term T of (<^, /)^' involves fewer than m — 1 + 1 symbols a, ^, . . . , and hence is 
of degree < m. 



90 ALGEBRAIC INVARIANTS 

of which the first two may be discarded as before. It remains 
to consider (/,/)*, for ^ = 1, 2, 3. By § 52, 

= (a/3)2(a7) lKi35)7x-5.2 + |(y5)i3.7x5/|. 

Replacing {P8)yz by iy8)^s + il3y)8„ and noting that 

{amay){^yH5.' = {H,f)--f=0, 
we get 

(/,/) = (a/3)2(a7)(75)/^.7x5/. 

Interchange y and 5, Hence 

U,/)=K«/3)'(75))3.7x5xK«7)5x + (5a)7x}. 
The quantity in brackets equals —{yd)ax. Hence 

Denoting H by Jh^ = h"^i, we have 



/ = (//,2^^^3) = (/,^)/,^^^2^ / = ^: 



X 7 



by the theorem in § 60. Here J = {haYax = {H, f)^ = 0. Since 
the first term is changed in sign when a and /3 are interchanged, 
we have {J,fy = 0. 
For the third case, 

(/, /)3 = (M)2(a7)^x7x2, 5.3)3 = {amay)(^8){y8y = D, 

an invariant, evidently equal to {H, H)^, the discriminant 
of H. Thus D is the discriminant of / (§§8, 30) and is not 
identically zero. Hence D is the only irreducible covariant 
of degree four. 

We can now prove by induction that /, H, J and D form 
a complete set of irreducible covariants of degree ^ ;w ^ 5. Let 
this be true for covariants Cm-\ of degree ^w — 1. We may 
discard (C^-i, fy if Cm-i has the factor/ or /, each of which 
is of order 3 ^ ^ and of degree (1 or 3) less than m — 1; and 
evidently also if it has the factor D. Hence Cm-\=n% e ^ 2. 
If ^ J 2, it has the factor E of order 2^k and degree 2<m — l. 
It remains to consider {E% f)^. If e>2, H^ has the factor 



§ 61] LAW OF RECIPROCITY 91 

H' of order 4^3 and degree -Kw— 1, since H'^ is of degree 
^ 6. Finally, 

{H\ / )3 = {hVl\, a/)3 = {ha)Hh'a)h\ = {h\ (JlaY-a;) = 0. 

Hence /, H, J, D form a fundamental system of covariants 
{cf. §30). 

63. Higher Binary Forms. The concepts introduced by 
Gordan in his proof of the liniteness of the fundamental system 
of covariants of the binary p-ic enabled him to find * the 
system of 23 forms for the quintic, the system of 2G forms for 
the sextic, as well as to obtain in a few lines the system for 
the cubic (§ 62) and the quartic (§ 31). Fundamental sys- 
tems for the binary forms of orders 7 and 8 have been deter- 
mined by von Gall.f 

Gordan's method yields a set of covariants in terms of 
which all of the covariants are expressible rationally and 
integrally, but does not show that a smaller set would not 
serve similarly. The method is supplemented by Cayley's 
theory | of generating functions, which gives a lower limit 
to the number of covariants in a fundamental system. 

64. Hermite's Law of Reciprocity. This law (§ 27) can be 
made self-evident by use of the symbolic notation. Let the 
form 

<t)=ax^ = ^x^ = - . .=ao{xi — piX2){xi— P2X2) . . . {xi — ppX2) 

have a covariant of degree d, 

K = ao'^^ipi-p-2)Kpi-p-syip2-P3)^ . . . (xi-pi-rs)'' . . . (xi-ppa:.)^ 

so that each of the roots pi, . . . , pp occurs exactly d times 
in each product. Consider the binary ^-ic 

J = ai'^ = b/ = . . . = CoCti— riXo) . . . {xi — raXo). 

* Gordan, Inmriantentheorie, vol. 2 (1887), p. 236, p. 275. Cf. Grace and 
Young, Algebra of Invariants, 1903, p. 122, p. 128, p. 150. 

t M alhematische Annaleii, vol. 17 (1880), vol. 31 (18S8). 

X For an introduction to it, see Elliott, Algebra of Quantics, 1895, p. 165, p. 
247. 



92 ALGEBRAIC INVARIANTS 

To the various powers, whose product is any one term of K, 

{pi-p'zY, {pi-p-j.y, {p2-p-iY, . . ., 

(X1-P1X2)'', {xi-p2X2)^, . . ., 

we make correspond the symboUc factors 

{ahy, {acy, (be)'', . . ., aj\ h\ . . . 

of the corresponding covariant of/: 

C={ahy{acy{bcY . . . aj'hjx'j' . . ., 

of degree p (since there are p symbols a, b, c, . . . , cor- 
responding to pi, . . . , Pj,) and having the same order 
/i+^2+/3 + . . . as A'. Conversely, C determines A. 

EXAMPLES 

Let p = 2. To K = au^^(pi — Pi)^'' corresponds the invariant C= (ai)^^ 
of degree 2 of f=(ix'^ = hx'^^. Again, to the covariant K4>^ of (/> corresponds 
the covariant (ab)'^^ OxVV of the form ax'^'^'^^ = bx^^'^K 

Concomitants of Ternary Forms in Symbolic Notation, 

§§ 65-67 

65. Ternary Form in Symbolic Notation. The general 
ternary form is 

'' r\s\l\ 

where the summation extends over all sets of integers r, 5, /, 
each = 0, for which r-\-s-\-l = n. 
We represent/ symbohcally by 

/ = a_," = |3/ . . . , ax = a\X\-^a2X2-\-OL2,Xz, .... 

Only polynomials in ai, 02, "a of total degree n have an inter- 
pretation and 

Just as a\^2—(X2^\ was denoted by (a/S) in §39, we now 
write 

(a/37) 



a\ 


0:2 


OL-i 


^1 


/32 


/33 


71 


72 


73 



§66] CONCOMITANTS OF TERNARY FORMS 93 

Under any ternary linear transformation 
T: ^i=^tXi + 7jtX2 + f«X3 (j = l,2, 3) 

ax becomes Q:{Xi+a,X2+«5-A'3, and/ becomes 

fi ! 
Z , '. ATstX^\X''2X^-s = (aiXi-\-a^X2-\-oiiX:i) . 
risltl 

Thus ax behaves Uke a covariant of index zero of /. Also 

«{ a, «{• 

^j /3, ^f =M7J(^r7f), 
7£ 7., 7r 

so that (aj37) behaves like an invariant of index unity of/. 

EXERCISES 

1. The discriminant of a ternary quadratic form ax^ is ^ {aftyY. 

2. The Jacobian of a^', Px"^, tz" is /ww (a/3TJa:c^~Vi'"~V/~^ 

3. The Hessian of a^" is the product of (a/37)2a/~^/3z"~^7i"~^ by a 
constant. 

4. A ternary cubic form ai' = /3i' = . . . has the invariants 

{a0y){a0b){ayb){fiyb), (a0y)(a0o)(aye)(0y<}>)(oe<t>y. 

66. Concomitants of Ternary Forms. If ui, ii2, uz are 
constants, 

Uz = UiXi-\-U2X2+U:iX3 = 

represents a straight line in the point-coordinates xi, xo, X3. 
Since ui, U2, M3 determine this line, they are called its line- 
coordinates. If we give fixed values to xi, X2, X3 and let the 
line-coordinates ui, U2, ws take all sets of values for which 
Ux = 0, we obtain an infinite set of straight lines through the 
point (xi, X2, X3). Thus, for fixed x's, Ux = is the equation 
of the point (xi, X2, x^) in line-coordinates. 

Under the linear transformation T, of § Go, whose deter- 
minant {^vO is not zero, the line Ux = is replaced by 

Ux=UiXi-{-U2X2 + U3X3=0, 

in which 333 

f/i = S ^iUi, 1/2= ^ ViUi, t/3 = - fi«<- 

t=l i=l 1=1 



94 ALGEBRAIC INVARIANTS 

The equations obtained by solving these define a linear trans- 
formation Ti which expresses wi, U2, us as linear functions 
of Ui, U2, Us and which is uniquely determined * by the 
transformation T. Two sets of variables Xi, X2, X3 and wi, ti2, U3, 
transformed in this manner, are called contragredient. 

A polynomial P{c, x, u) in the two sets of contragredient 
variables and the coefficients c of certain forms Ji{x\, X2, xz) 
is called a mixed concomitant of index \ of the /'s if, for every 
linear transformation T of determinant A?^0 on xi, X2, X3 and 
the above defined transformation Ti on ui, U2, uz, the product 
of P{c, X, u) by a'' equals the same polynomial P(C, A', U) 
in the new variables and coefficients C of the forms derived 
from the /'s by the first transformation. For example, Ux is 
a concomitant of index zero of any set of forms. 

In particular, if P does not involve the w's, it is a covariant 
(or invariant) of the /'s. If it involves the u's, but not the 
re's, it is called a contravariant of the /'s. 

Since U\=u^, U2 = u^, 1/3=11^, we see by the last formula 
in § 65, with 7 replaced by u, that (a^u) behaves like a contra- 
variant of index unity of ax^, and also like one of ax", ^x^. 

For the linear forms ax and fix, («/3«) has an actual interpretation. 
For /=«!= = /So:-, where 

/= a2ooXr +ao2oX2^ +0002X3^ +2auoXiX2 +20101X1X3 +2aonX2X3, 
it may be shown that 



O200 duo ClOl 111 

Olio flo20 floil Ui 

Gioi flon aoo2 ^h 

III Ui Hi 



{afiuY 



By equating to zero this determinant (the bordered discriminant of 
/), we obtain the line equation of the conic /= 0. 

67. Theorem. Every concomitant oj a system of ternary forms 
is a polynomial in Ux and expressions of the types ax,{a^y), {a^u). 

* We have only to interchange the rows and columns in the matrix of T and 
then take the inverse of the new matrix to obtain the matrix of the transforma- 
tion Ti. Similarly, Xu x-i are contragredient with Ui, Ui, if we have T, § 40, and 

«i = (^72 Ui - & Ui) / (It?) , Ui = ( - ,,1 t/i+ ti Ui) I ( i-n) . 



167] 



CONCOMITANTS OF TERNARY FORMS 



95 



a 


a 
a ^2 


a 
a ^3 




a 


_a_ 

dm 


_a_ 


a^ 


9171 


a^ia'72ar3 


a 
afi 


a 
af2 


a 
ars 





A concomitant of the forms /<(xi, X2, X3) is evidently a 
covariant of the enlarged system of forms /i and Ux. We may 
therefore restrict attention to covariants. In the proof of the 
corresponding theorem for binary forms, we used the operator 
(1), § 42. Here we employ an operator V composed of six terms 
each a partial differentiation of the third order: 



V 



the determinant being s>Tnbolic. It may be shown as in 
§ 43 that 

F(^77f)" = «(» + l)(w + 2)(|r?f)"-l. 

As in § 44, the result of applying F'" to a product of k factors 
of the type «{, / factors of the type /3„ and m factors of the 
type 7J-, is a sum of terms each containing k—r factors a^, 
l—r factors /3„ m—r factors 75-, and r factors of the type (a/37). 
For the case of an invariant /, the theorem can be proved 
without a device. In the notations of § 65, we have 

/U) = (^,f)V(a). 

Each yl is a product of factors aj, a,, a^. Hence I {A) equals 
a sum of terms each with X factors of the type a^, X of t}'pe 
a„ and X of type a^. Operate on each member of the equation 
with F^. The left member becomes a sum of terms each a 
product of a constant and factors of t}^e (ajSy). The right 
member becomes the product of 7(a) by a number not zero. 
Hence I equals a polynomial in the (a^y). 
For a covariant K, we have, by definition, " 

K(A,X)=(^rji)^K{a,x). 

Solving the equations of our transformation T in § 65, 
we get 

(|77i')A'i =a;i(7?2S''3 — ^3^2) +^"2(773^1 — •71^3) -}-X3{r]i^2 — 772ri), 



96 ALGEBRAIC INVARIANTS 

etc. Replacing xi by y2Z3-ys^2, X2 by yaZi-yiSs, and xs by 
yiZ2—y2Zi, we get 

{^vOXi^yr,z^-y^Zr„ 

{h^)X2 = ycZ^-yiZ^, 

(^v^)X3=yiZ,-y„Zi. 

Our relation for a covariant K of order co now becomes 

S(product of factors a^, y^, z^, a,,, . . . , z^) = {^r]^Y'^''K{a, x) . 

each term on the left having \+co factors with the subscript ^, 
etc. Apply the operator V to the left member. We obtain 
a sum of terms with one determinantal factor (a/37), ("i3y) or 
{ayz)=ax, and with X+co — 1 factors with the subscript |, etc. 
The result may be modified so that the undesired factor (a^y) 
shall not occur. For, it must have arisen by applying V to 
a term with a factor Hke a^^^y^ and hence (by the formulas 
for the Xi) with a further factor s, or z^. Consider therefore 
the term Ca^jS^y^z, in the initial result. Then the term 
— Caj|8,>\Sf must occur. By operating on these with F, 
we get C{a^y)Zr„ — C(a/3s)y„ respectively, whose sum equals 

CI (/3y2K - (ays)^,N C(/3.a, -ax/3,) , 

as shown by expanding, according to the elements of the last 
row. 

Oil /3i yi zi 

a2 /32 )'2 Z2 
«3 /33 ys Z3 

Oir, ^„ ytl Zr, 



:0. 



The modified result is therefore a sum of terms each with 
one factor of type (ai37) or ax and with X+oj — 1 factors with 
subscript ^, etc. 

Applying V in succession X + co times and modifying the 
result at each step as before, we obtain as a new left member 
a sum of terms each with \-\-w factors of the types {al3y) and 
ax only. From the right member we obtain nK, where n is 
a number ?^0. Hence the theorem is proved. 



§68] QUATERNARY FORMS 97 

68. Quaternary Forms. For ax=Q;i:ri+. . .-{-a^Xi, 

{ „ n an », n tn 

J —OLx = Pi — 7x — Ox 

has the determinant {a^yb) of order 4 as a symbolic invariant 
of index unity. Any invariant of / can be expressed as a 
polynomial in such determinantal factors; any co variant as 
a polynomial in them and factors of type ax. In the equation 
«x = of a plane, n\, . . ., W4 are called plane-coordinates. 
The mLxed concomitants defined as in § 66 are expressible 
in terms of Ux and factors hke ax, (a^y d) , ia^yu) . For geometrical 
reasons, we extend that definition of mked concomitants to 
polynomials P{c, x, u, v), where Vi, . . . , 1)4 as well as wi, . . . , W4 
are contragredient to xi, . • . , X4.' There may now occur 
the additional type of factor 

{a^Uv) = {ai^2 -Ot2^l) (usVi-UiV^) +. . . -f- (0:3/34 -0:4183) {uiV2 -U2V1). 

These six combinations of the w's and v's are called the line- 
coordinates of the intersection of the planes Wx = 0, Vx = 0. For 
instance, {a^uv)~ = is the condition that this line of inter- 
section shall touch the quadric surface ax^ = 0. 

We have not considered concomitants involving also a 
third set of variables wi, . . • , w^, contragredient with the x's. 
For, in 

Mi:»:i-|-. . .-\-u^Xi = 0, z'lXi-f-. . .-\-V4X4, = 0, 

WiXi-}-. . .+^4^4 = 0, 

xi, . . . , Xi are proportional to the three-rowed determinants 
of the matrix of coefficients, so that (auvw) is essentially a*. 



INDEX 



(The numbers refer to pages) 



Absolute invariant, 51 
Alternants, 41 
Annihilators, 34, 39, 72 
Apolarity, 78-84 

Binary form, 14, 91 

Canonical form of cubic, 17 

quartic, 50 

ternary cubic, 28 

Concomitants, 93, 97 
Conic, 2, 21, 24, 94 
Contragredient, 94 
Contravariant, 94 
Convolution, 85 
Covariant, 12, 15, 66 

— in terms of roots, 56 
symbolic factors, 67, 95 

— as invariant, 71 
Cross-ratio, 5, 15, 56 
Cubic curves, 25-29, 81 

— form, 14, 16, 48, 80, 89, 93 

Degree, 30 

Differential operators, 36, 59, 95 

Discriminant of binary cubic, 17, 36 

quadratic, 10 

p-\c, 55 

ternary quadratic, 24 

Double point, 83 

Euler's theorem, 15, 41 

Finiteness of covariants, 70-76 

syzygies, 76 

Forms, 14 

Functional determinant, 12 

Fundamental system, 48, 61, 84-91 



Harmonic, 15, 20, 78 

Hermite's law of reciprocity, 45, 91 

Hessian, 11, 15-18, 23-28, 58, 66, 84, 93 

— curve, 25 
Hubert's theorem, 72 
Homogeneity, 14, 30, 37 
Homogeneous coordinates, 8, 20 

Identity transformation, 33 
Index, 10, 14, 15, 31, 32 
Inflexion point, 26-28, 82 

— tangent, 26, 82 

— triangle, 27 
Intermediate invariant, 19 
Interpretation of invariants, 2, 10, 23 
Invariant. 1, 10, 14, 28 

— in terms of roots, 54 
Inverse transformation, 33 
Irrational invariant, 55 
Irreducible covariant. 87 
Isobaric, 31, 32, 38, 42 

Jacobian, 12, 15, 18. 29, 65, 83. 93 

Leader of covariant. 40. 43, 58 
Line coordinates. 93, 97 

— equation of conic. 94 
Linear form, 9, 14, 33. 84 

— fractional transformation, 6, 22 

— transformation, 3, 9, 22. 33. 34, 

Mixed concomitant, 94, 97 

Order, 14 

Partitions. 44, 45 
Perspective, 4 

99 



100 



INDEX 



Plane coordinates, 97 
Product of transformations, 33 
Projective, 4, 23 
— property, 10, 11,23 
Projectivity, 5, 6 

Quadratic form, 10, 14, 48, 84 
Quartic, 14, 36, 49, 83 
Quaternary form, 97 

Range of points, 4 
Rational curves, 81 
Reciprocity. See Hermite. 
Resultant, 10, 18, 19 



Seminvariant, 40, 42-50, 64 
— in terms of roots, 53 
Singular point, 25 
Solution of cubic, 17 

quartic, 52 

Symbolic notation, 63 
Syzygy, 49, 50, 76 

Ternary form, 14, 24, 25, 92 
Transformation. See Linear. 
Transvectants, 77, 85 

Unary form, 14 

Weight, 31, 32, 38 



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